UC-NRLF 


*B    E&Q    M7S 


WIS  SLOWS 


RCIA1  CALCULATOR. 


/ 


Digitized  by  the  Internet  Archive 

in  2007  with  funding  from 

Microsoft  Corporation 


http://www.archive.org/details/foreigndomesticcOOwinsrich 


THE 


FOREIGN   AND    DOMESTIC 


COMMERCIAL    CALCULATOR ; 


OR, 


A   COMPLETE   LIBRARY   OF  NUMERICAL,  ARITHMETICAL,  AND 

MATHEMATICAL  FACTS,  TABLES,  DATA,  FORMULAS,  AND 

PRACTICAL    RULES    FOR    THE    MERCHANT    AND 

MERCANTILE  ACCOUNTANT. 


BY 

E.    S.    WINSLOW. 

Author  of  "Comprehensive  Mathematics,"  " Computists'  Manual," 

"  Machinists'  and  Mechanics'  Practical  Calculator  and  Guide," 

"  Tin-plate  and  Sheet-iron  Workers'  Monitor." 


Fourth    Edition.     JEnlarped. 


BOSTON: 
PUBLISHED    BY    THE    AUTHOE. 

186  7. 


HFS 


GEO.  W.  LINDSEY,  Local  Agent  for  the  sale  of  Winslow's  Mathematical 
Works,  No.  897  Washington  Street,  Boston. 


Entered,  according  to  Act  of  Congress,  in  the  year  1867,  by 

E.    S.    WIN  SLOW, 

In  the  Clerk's  Office  of  the  District  Court  of  the  District  of  Massachusetts. 


Stkekottpkd  bt  C.  J.  Pktkbs  &  So*. 


Printed  by  Wm.  A.  11*11,  No.  40,  CongreM  Street,  Bo»ton. 


PREFACE 

TO   THE   COMPEEHENSIVE,  MATHEMATICS. 


On  presenting  this  work  to  the  public,  it  may  be  proper  to  state 
that  it  has  been  designed  and  written  mainly  for  the  practical 
man.  It  contains  a  vast  array  of  Numerical,  Arithmetical,  and 
Mathematical  facts,  tables,  data,  formulas,  and  rules,  pertaining 
to  a  great  variety  of  subjects,  and  applicable  to  a  diversity  of  ends, 
as  well  as  much  information  of  a  more  general  nature,  valuable  to 
the  artisan,  and  commercial  classes ;  thus  meeting  the  wants,  in  an 
eminent  degree,  of  the  lovers  of  the  exact  sciences,  and  the  prac- 
tical wants  of  students  in  the  mathematics. 

The  facts  and  data  alluded  to  have  been  gathered,  with  much 
care  and  patience,  from  a  great  variety  of  sources,  or  derived,  often 
by  toilsome  investigations,  from  known  and  accredited  truths. 
The  care  that  has  been  taken  in  respect  to  these,  it  is  thought, 
should  secure  for  this  particular  department  reliance  and  trust. 

The  tables,  which  are  numerous,  have,  with  few  exceptions,  been 
composed  and  arranged  expressly  for  the  work,  and  a  confidence 
is  felt  that  they  may  be  relied  on  for  accuracy. 

From  the  valuable  works  of  Dr.  Ure,  Adcock,  Gregory,  Grier, 
Brunton ;  from  the  publications  of  the  transactions  of  London, 
Edinburgh,  and  Dublin  Philosophical  Societies ;  and  from  the  pub- 
lications by  the  Smithsonian  Institute,  much  valuable  information 
lias  been  gained,  relating  mainly  to  machinery  and  the  arts  ;  and 
to  these  sources  the  author  feels  indebted. 

The  conciseness  with  which  the  work  has  been  generally  written 
would,  perhaps,  be  found  an  objection,  were  it  not  that  all  the  pro- 
positions and  problems  of  intricacy  are  accompanied  with  exam- 
ples and  illustrations,  and,  in  the  matters  of  Geometry,  additionally 
accompanied  with  diagrams.  The  whole,  it  is  thought,  will  appear 
clear  to  him  who  consults  it.  A  prominent  feature  in  the  design 
has  been  to  produce  a  useful  work,  and  one  which  in  the  way  of 
price«hall  be  readily  accessible  to  all. 

284302 


PREFACE. 

PREFA  CE 

TO   THE   FOREIGN   AND    DOMESTIC    COMMERCIAL 
CALCULATOR. 


This  work  is  composed  of  the  first  four  sections  of  the  author's 
"  Comprehensive  Mathematics."  It  was  thought  advisable 
to  publish  this  portion  of  that  work  in  a  separate  form  on  account 
of  price ;  more  especially  as  it  contains  all  of  a  commercial  nature 
treated  of  in  that  work.  Indeed,  the  contents  of  that  work  were 
arranged  expressly  to  this  end.  The  Table  of  Contents  in  both 
works  is  the  same.  The  work  being  stereotyped,  this  could  not  well 
be  avoided.  The  Table  of  Contents,  therefore,  in  either  work,  is 
that  of  the  "  Comprehensive  Mathematics,"  and  the  first 
four  sections  thereof;  that  is,  Section  I.,  Section  II.,  Section  III., 
and  Section  A.,  is  that  of  the  "  Foreign  and  Domestic  Commer- 
cial Calculator." 


PREFA  CE 
TO  THE  TIN-PLATE  AND  SHEET-IRON  WORKERS5  MONITOR. 


This  work  is  composed  of  Section  VI.  of  the  author's  "  Com- 
prehensive Mathematics,"  with  portions  of  other  sections  of 
that  work.  It  embraces  all  that  is  contained  in  the  last-mentioned 
work  of  special  interest  to  the  Tinsmith,  as  such.  It  may  be  re- 
lied on  for  accuracy  in  all  particulars,  and  is  believed  to  bo  tlu* 
first  and  only  reliable  work  of  the  kind  ever  published.  It  is  pub- 
lished in  separate  form  on  account  of  price,  and  with  the  view  of 
affording  apprentices  and  students  every  possible  facility  of  obtain- 
ing it.  It  contains  over  100  pages,  nearly  50  diagrams,  and  step- 
by-step  directions  for  constructing,  mechanically,  not  less  than  BO 
unlike  and  diiK  rent  patterns,  embracing  all  of  the  more  difficult 
and  complicated  in  use,  and  several  of  new  and  beautiful  designs. 


CONTENTS. 


SECTION    A. 

PAGE 

Foreign  Moneys  of  Account ...  a    1 
Foreign  Linear  and  Surface  Meas- 
ures   a  14 

Foreign  Weights •  «8 

Foreign  Liquid  Measures  ....  a  35 

Foreign  Dry  Measures a  44 

Custom  House  Allowances  on  Du- 
tiable Goods,  &c a  51 

Table  of  Established  Tares   .  .  .a  52 


SECTION    I. 

MONEYS  OF  ACCOUNT,  COINS, 
WEIGHTS,  AND  MEASURES  OF 
THE  UNITED  STATES;  FOREIGN 
GOLD   COINS,  &C. 


Explanations  of  Signs 


12 


Moneys  of  Account  of  the  United 
States 13 

Comparative   Value  of    Gold  and 
Silver ■  .    .   .    13 

Gold,  pure ;  value  of,  by  weight .  .    15 
Mint  Gold,  Standard  of,  &c.    .  .   .    15 
Gold  Coins,  their  weights  and  val- 
ues   15 

Silver,  pure;  value  of,  by  weight .  10 
Mint  Silver,  Standard  of,  &c.  .  .  .  16 
Silver    Coins,   their   weights    and 

values 10 

Copper  Coins,  &c 10 

Present  Far  Value  of  Silver  Coins 

issued  prior  to  June,  1853  ....    17 
Currencies  of  the  different  States 

of  the  Union 17 

The  Metrical    System  of  Weights 

and  Measures 18 

Foreign  Gold  Coins,  Tables  of,  &c.  19 
Foreign   Silver   Coins,  Values  of,  25 

1* 


WEIGHTS  AND  MEASURES. 

PACK 

Long  or  Linear  Measure  ...  25 

Cloth  Measure 25 

Land  Measure 25 

Engineer's  Chain 25 

Shoemaker's  Measure 26 

Miscellaneous  Measures 26 

Square  or  Superficial  Meas- 

uhe 26 

Measure  for  Land 26 

Circular  Measure 27 

Cubic  or  Solid  Measure  ...  27 
General  Measure  of  Weight,  28 

Gross  Weight 28 

Troy  Weight 28 

Apothecaries1  Weight, 28 

Diamonds,  Measure  of  Value,  &c,  28 

Liquid  Measure 28 

Imperial  Liquid  Measure 29 

Ale  3Ieasure    29 

Dry  Measure 29 

Imperial  Dry  Measure 30 


SECTION    II. 

MISCELLANEOUS  facts,  calcu- 
lations, AND  MATHEMATICAL 
DATA. 

Specific  Gravities,  Tables  of,  31 

Weight  per  Bushel  of  Articles  .  .  35 
Weight  per  Barrel  of  Articles  ...  35 
Weights  of  different  Measures  of 

various  Articles 35 

Weight  of  Coals,  &c,  Tables  .  35,  55 
Practical  Approximate  Weight  in 

Pounds  of  Various  Articles  ...   36 


Ropes  and  Cables  . 
v 


CONTENTS. 


PAOB 

Weight  and  Strength  of  Iron 
Chains 37 

Comparative  Weight  of  Metals, 
Table 38 

Weight  of  Rolled  Iron,  Square  Bar, 
Tables 38 

Weight  of  Various  Metals,  differ- 
ent Forms  of  Bar 39 

Weight  of  Bound-rolled  Iron,  Ta- 
bus   40 

Weight  of  Cast-iron  Prisms  of  dif- 
ferent forms,  &c 40 

Weight  of  Flat-rolled  Iron,  Table,  42 

Weight  of  Different  Metals, in  Plate,  44 

The  American  Wire  Gauge  .  45 

The  Values  of  the  Nos.  American 
Wire  Gauge  and  Birmingham 
Wire  Gauge,  in  the  United  States, 

inch,  Tables  of 45 

The   Number  of  Linear  Feet  in  a 
Found  of  different  kinds  of  Wire 
of  different  Sizes,  Table  of,  &c,  46 
Characteristics.  &c,  of  Alloys  of 

Copper und Zinc,— Brass.  ...   47 
The  Weight  per  Square  Foot  of  dif- 
ferent Boiled  Metals  of  different 
thicknesses  by  the  Wire  Gauge, 

Table 48 

Tin  Plates,  Sizes,  &c,  Table,  vj 
.Sheet    Iron,    Sheet    Zinc,    Copper 
Sheathing,  Yellow  Metal,  Weight 

of,  &c 49 

Capacity  in  Gallons  of  Cylindrical 

Cans,  &c,  Table 50 

Weight  of  Pipes 52 

Weight  of  Pipes,  Table 53 

Weight   of    Cast-iron     and   Lead 

Balls 54 

Weight  of  Hollow  Balls  or  Shells,  54 

Analysis  of  Coals 55 

Weight,    Heating  Power,   Ac,  of 
Is  and  other  kinds  of  Fuel, 

Tabu 55 

iiArioN  of  Lumber  .  .  .  5G 

Hoard   Measure 5f, 

T6  Measure  Square  Timber .  ...  5(5 
-me  Round  Timber ....  50 
Tab  lb  relative  to  the  Measurement 

of  Round  Timber 57 

To  flind  the  Solidity  of  the  gr< 

ingular  stick  that  can  be  cut 

from  a  Log  of  Given  Dimensions,  58 

To  Bud  the  solidity  of  the  greatest 

ire  St  irk  that  can  be  oat  from 

a  Hound  Stick    of  Given    Dimen- 



To  find  the  Contenti  oi  ■  Login 

Hoard  Measure 69 

o GO 


paos 

To  find  the  Dimensions  of  Vessels 
of  different  Forms,  for  holding 
Given  Quantities 62 

Cask  Gauging,  all  Forms  of 
Casks 63 

To  find  the  Contents  of  a  Cask,  the 
same  as  would  be  given  by  the 
Gauging  Bod 66 

To  find  the  Diagonal  and  Length 
of  a  Cask 66 

Ullage 07 

To  find  the  Ullage  of  a  Standing 
Cask 67 

To  find  the  Ullage  when  the  Cask  is 
upon  its  Bilge 67 

To  find  the  Quantity  of  Liquor  in  a 
Cask  by  its  Weight 68 

Customary  Bide  by  Freighting  Mer- 
chants for  finding  the  Cubic 
Measurement  of  Casks 68 

Tonnage  ok  Vessels,  to  Calcu- 
late   69 

Of  Conduits,  or  Pipes 70 

To  find  the  requisite  thickness  of  a 
Pipe  to  support  a  Given  Head  of 
Water 70 

To  find  the  Velocity  of  Water  pass- 
ing through  a  Pipe   71 

To  find  the  Head  of  Water  requi- 
site to  a  Bequired  Velocity 
through  a  Pipe 71 

To  find  the  Quantity  of  Water  Dis- 
charged by  a  Pipe  in  a  Given 
Time ,    71 

To  find  the  Specific  Gravity  of  a 
Hodv  heavier  than  Water  ....    72 

To  find  the  Specific  Gravity  of  a 
Body  lighter  than  Water  ....    72 

To  find  tne  Specific  Gravity  of  a 
Thud 72 

To  find  the  Quantity  of  each  of  the 
several  Metals  composing  an  AI- 
lov 72 

To  find  the  Lifting-power  of  a  Bal- 
loon   73 

To  find  the  Diameter  of  a  Halloon 
equal  to  the  Raising  of  :•  Given 
Weight 73 

To  find  the  Thickness  of  ■  iMlow 
Metallic  Globe  that  shall  bai 
Given    Buoyancy    in    a    (liven 
Liquid 73 

To  Cut  a  Square  Sheet  of  Metal  bo 
as  to  form  a  Vessel  of  the  Great- 
est Capacity  the  Sheet  admits  of.   73 

Comparative  cohesive  Forces  of 
Substances,  Table 74 

Alloys  having  a  Tenacity  greater 
than  the  Sum  of  their  Con- 
stituents     71 


CONTENTS. 


Vll 


PAOB 

Alloys  having  a  Density  greater 
than  the  Mean  of  their  Con- 
stituents     75 

Alloys  having  a  Density  less  than 
the  Mean  Of  their  Constituents  .    75 

Relative  Powers  of  different  Metals 
to  Conduct  Electricity 75 

Dilations  of  Solids  Imf  Seat,  Table  7S 

Melting  Points  of  Metals  and  other 
Substances,  TABLE 7G 

Relative  Powers 'of  Substances  to 
Radiate  Heat,  Table 76 

Dolling  Points  of  Fluids 76 

Freezing  Points  of  Fluids  ....  77 
Expansion  of  Fluids  by  Heat .  .  .  77 
Relative  Powers  of  Substances  to 

Conduct    Heat 77 

Ductility  and  .Ualeability  of  Metals,  77 
Quantity    per  cent,   of   Nutritious 
Matter  contained  in  different  Ar- 
ticles of  Food 78 

Standard,  &*.,  of  Alcohol 78 

Quantity  per  cent,  of  Absolute  Al- 
cohol contained  in  different  Pure 
Liquors,  Wines,  &C.,  TABLE .  .  7S 
Proof  of  Spirituous  Liquors  ...  78 
Comparative  Weight  ot  Timber  in 
a  Green  and  Seasoned  State,  TA- 
BLE, &c 79 

Relative  Power  of  different  kinds  of 

Fuel  to  Produce  Heat,  TABLE,  .    79 
Relative  illuminating  Power  of  dif- 
ferent Materials,  Table  and  Re- 
marks,     80 

Thermometers,  different  kinds, 
to  Reduce  one  to  another,  &c, .   82 

Horse-Power 83 

Animal  Power 83 

Steam,   Tables   in  relation  to, 

&c, &},  308 

Velocity  and  Force  of  Wind,  Ta- 

BLE 84 

Curvature  of  the  Earth  .  .  .  .  84,  213 
Degrees  of  Longitude,  Lengths  of, 

&c 84 

Time,  with  respect  to  Longitude,  84 

Velocity  of  Sound 84 

Velocity  of  Light 85 

Gravitation 85,302 

Area  of  the  Earth,  its  Density,  &c,  85 

Chemical  Elements 80 

Elementary  Constituents  of  Bodies, 

Table  .  .  .  .  - 87 

Combinations  by  Weight  of  the 
Gases    in   forming   Compounds, 

Table 87 

Combinations  by  Volume  of  the 
Gases,  their  Condensation,   &c, 

in  forming  Compounds 88 

Atomic  Weight «9 


PAOB 

CImmical  and  other  Properties  of 
Various  Substances 90 


SECTION    III. 


practical  arithmetic. 

Vulgar  Fractions 95 

Reduction  of  Vulgar  Fractions  .  .  95 
Addition  of  Vulgar  Fractions  .  .  .  09 
Subtraction  of  Vulgar  Fractions  .  99 
Division  of  Vulgar  Fractions  .  .  .  100 
Multiplication  of  Vulgar  Fractions  100 
Multiplication     and    Division     of 

Fractions  Combined 101 

Cancellation 90,  97, 102 

To  Reduce  a  Fraction  in  a  higber, 
to  an  equivalent  in  a  given  low- 
er denomination 102 

To  Reduce  a  Fraction  in  a  lower, 
to  an  equivalent  in  a  given  high- 
er denomination 102 

To  Reduce  a  Fraction  to  Whole 
Numbers  in  lower  given  denom- 
inations      103 

To  Reduce  Fractions  in  lower  de- 
nominations to  given  higher  de- 
nominations     103 

To  work  Vulgar  Fractions  by  the 
Rule  of  Three,  or  Proportion  .  .  104 

Decimal  Fractions 104 

Addition  of  Decimals 105 

Subtraction  of  Decimals 105 

Multiplication  of  Decimals  ....  106 

Division  of  Decimals 106 

Reduction  of  Decimals 107 

To  work  Decimals  by  the  Rule  of 

Three 108 

Proportion,  or  Rule  of  Three  ...  109 

Compound  Proportion 110 

Conjoined   Proportion,  or  Chain 

Rule 112 

Percentage 114 

interest 120 

Compound  Interest 122 

Bank  Interest,  or  Bank  Discount .  127 

Discount 129 

Compound  Discount 129 

Profit  and  Loss 130 

Equation  of  Payments 132 

General  Average 134 

Assessment  of  Taxes 136 

Insurance 136 

Life  Insurance  • 136 

Fellowship 138 


Vlll 


CONTENTS. 


TkOX 

Alligation 139 

Involution 141 

Evolution 141 

To  Extract  the  Square  Root    ...  142 
To  Extract  the  Cube  Root    ....  143 

To  Extract  any  Root 145 

Arithmetical  Progression 146 

Geometrical  Progression 150 

Annuities 154 

Of  Installments  generally    .  .  .  .  1G4 

Permutation 1GG 

combination 167 

Problems 169 


SECTION    IV. 

GEOMETR*. 

Definitions,  Construction  of 
Figures,  &c 172 

To  Bisect  a  Line 176 

To  Erect  a  Perpendicular 176 

To  Let  Fall  a  Perpendicular    .  .  .  176 

To  Erect  a  Perpendicular  on  the 
end  of  a  Line 177 

To  draw  a  Circle  through  any  three 
points  not  in  a  straight  line,  and 
to  find  the  Centre  of  a  Circle,  or 
Arc 177 

To  find  the  Length  of  an  Arc  of  a 
a  Circle  approximately  by  Me- 
chanics   177 

From  a  given  Point  to  draw  a 
Tangent  to  a  Circle 177 

To  draw  from  or  to  the  Circumfer- 
ence of  a  Circle,  lines  tending 
to  the  Centre,  when  the  latter  is 
inaccessible 177 

To  describe  an  Oval  Arch  on  a 
given  Conjugate  Diameter   ...  178 

To  describe  an  Oval  of  a  given 
Length  and  Breadth 178 

To  describe  an  Arc  or  Segment  of 
a  Circle  of  Large  Radius  .  .  .  .179 

To  describe  an  Oval  Arch,  the 
Span  and  Ki.»e  being  given    .    .    .17!) 

Gothic  Arches,  to  draw 180 

Polygons,  to  oonstruot 181 

Polygons,  to  inscribe  in  a  given 

Circle 181 

Polygons,  to  circumscribe  about  a 

given  Circle 181 

To  produce  a  Square  of  the  mom 

Area  m  a  given  Triangle  .  .  .  -  181 
Mini  ;l  Parabola   ... 
•  a  Hyperbola    .  , 
To  biBect  auy  given  Triangle  .  .  .  1S2 


MM 

To  draw  a  Triangle  equal  in  Area 
to  two  given  Triangles 183 

To  describe  a  Circle  equal  in  Area 
to  two  given  Circles 183 

To  construct  a  Tothed,  or  Cog- 
wheel   183 

Of  the  Conic  Sections  .  .  .  .   1S4 

Mensuration  of  Lines  and  Super- 
ficies. 

Triangles 185 

Of  Right- Angled  Triangles    .  .    •  186 
Of  Oblique- Angled  Triangles    .  .    187 
To  find  the  Area  of  a  Triangle     .    188 
To  find  the  Hypotenuse  of  a  Tri- 
angle   189 

To  find  the  Base,  or  Perpendicu- 
lar, of  a  Triangle 188,  189 

To  find  the  Height  of  an  inacces- 
sible Object   189 

To  find  the  Distance  of  au  inac- 
cessible Object 190 

To  find  the  Area  of  a  Square, 
Rectangle,  Rhombus,  or  Rhom- 
boid   190 

To  find  the  Area  of  a  Trapezoid  .    191 
To  find  the  Area  of  a  Trapezium  .    191 
Of  Polygons,  Table,  &c.   ...   194 
To  find  the  Perpendieular  of  a 
Rhombus,  Rhomboid,  or  Trape- 
zoid   • 192 

To  find  the  Diagonal  of  a  Rhom- 
bus. Ithoniboid,  or  Trapezoid    .    192 
To  find  the  Area  of  a  regular  or 

irregular  Polygon 195 

Circle 196 

The  Circle  and  its  Sections  .  .  .  .197 
To  find  the  Diameter,  Circumfer- 
ence, and  Area  of  a  Circle  .   .   .    198 
To  find  the  Length  of  an  Arc  of  a 

Circle 199 

To  find  the  Area  of  a  Sector  of  a 

Circle 201 

To  find  the  Area  of  a  Segment  of 

I  I  irele 201 

To  find  the  Area  of  a  Zone     ...    202 
TO  And  the  Diameter  of  a  Circle  of 

Which  a  given  Zone  is  a  part  .   .    202 
To  And  the  Area  ofs  Crescent  .  .  202 
To  find  the  side  of  a  Square  that 
■hall  contain  an  Area  equal  to 
thai  Of  a  given  Circle 202 

To  And  the  Diameter  of  b  Circle 
that  shall  have  an  Ana  equal  to 
that  of  a  given  Square 202 

To  And  the  Diameters  of  three 
equal  circles  th<  that 

can  be  inscribed  in  a  given  ♦ir- 
ele  


CONTENTS. 


PAGE 

To  find  the  Diameters  of  four  equal 
circles  the  greatest  that  can  be 
inscribed  in  a  given  Circle  .  .  .  202 
To  find  the  Side  of  a  Square  in- 
scribed in  a  given  Circle  ....  203 
To  find  the  Diameter  of  a  Circle 
that  will  circumscribe  a  given 

Triangle 203 

To  find  the  Diameter  of  the  great- 
est Circle  that  can  be  inscribed 

in  a  given  Triangle 203 

To  divide  a  Circle  into  any  num- 
ber  of   Concentric   Circles    of 

equal  Areas 204 

To  find  the  Area  of  the  space  con- 
tained between  two  Concentric 

Circles  .  .  . 205 

Ellipse 205 

To  find  the  Area  of  an  Ellipse  .  .   20? 
To  find  the  Length  of  the  Circum- 
ference of  an  Ellipse  .  .....    207 

To  find  the  Area  of  an  Elliptic  Seg- 
ment     207 

Parabola 209 

To  find  the  Area  of  a  Parabola  .   .  210 
To  find  the  Area  of  a  Zone  of  a 

Parabola 210 

To  find  the  Altitude  of  a  Parabola,  210 
To  find  the  Length  of  a  Semi-para- 
bola    210 

Hyperbola 211 

To  find  the  Length  of  a  Semi- 
hyperbola  212 

To  find  the  Area  of  a  Hyperbola .  212 

Cycloid,  and  Epicycloid  ...  212 

To  find  the  Length  of  the  Curve  of 

a  Cycloid 213 

To  find  the  Area  of  a  Cycloid.  .  .   213 
To  find  the  Distance  of  Objects  at 

Sea,  &c 213 

Stereometry,  or  Mensuration 
of  Solids. 

Of  Prisms 214 

Of  Right  Prisms  or  Cubes  ....   215 
Of  Parallelopipedons  ....   215 

Of  Pyramids 215 

Of  Frustums  of  Pyramids   .  .  216 

Of  Prismoids 216 

Of  the  Wedge 217 

Of  Cylinders 217 

To  find  the  Length  of  a  Helix  .  .   217 

Of  Cones 218 

Of  Frustums  of  Cones.  .  .  .65,218 
Of  Spheres  or  Globes  ....  219 


PAGE 

Of  Spherical  Segments 219 

Of  Spherical  Zones 220 

To  find  the  greatest  Cube  that  can 
be  cut  from  a  given  Sphere  .  .  .   220 

Of  Spheroids    . 221 

Of  Segments  of  Spheroids  .  .  .  .221 
Of  the  Middle  Frustum  of  a  Spher- 
oid     65,221 

Of  Spindles 222 

Of  the  Middle  Frustum  of  a  Para- 
bolic Spindle 65,  222 

Of  Parabolic  Conoids  ...  65,  223 

Of  Hyperboloids 223 

To  find  the  Surface  of  a  Cylindri- 
cal Ring 224 

To  find  the  Solidity  of  a  Cylindri- 
cal Ring ' 224 

Of  the  regular  bodies  ....  225 
Promiscuous     Examples     in 

Geometry 226 

Trigonometry 231 

Tables  of  Sines,  Cosines,  Tan- 
gents, &c 241 

Tables   of    Squares,   Cubes, 
Square  and  Cube  Roots,  &c.  245 


SECTION    V. 

mechanical  powers,  mechani- 
cal centres,  circular  mo- 
tion, strength  of  materi- 
als; STEAM,  THE  STEAM  EN- 
GINE, ETC. 

The  Lever 271 

The  Wheel  and  Axle  ....  272 

The  Pulley 273 

The  Inclined  Plane 274 

The  Wedge 275 

TnE  Screw 275 

Transverse  Strength  of  Bodies    .   279 

Deflections  of  Shafts,  &c 286 

Resistance  of  Bodies  to  Tortion  .   287 
Resistance  of  Bodies  to  Compres- 
sion   289 

Centres  of  Surfaces 291 

Centres  of  Solids 293 

Centres  of  Oscillation  and 

Percussion 294 

Centre  of  Gyration 298 

Central  Forces 300 


CONTENTS. 


MM 

FltWheels 301 

The  Governor 301 

Force  of  Gravity 302 

To  find  the  Height  of  a  Stream 

projected  vertically  from  a  Pipe,  303 
To  find  the   Tower  requisite  to 

E reject  a  Stream  to  aiiy  given 

bight 303 

Of  Pendulums 304 

Screw-Cutting  in  a  Lathe  .  .  305 
Table    of    Change    Wheels    for 

Screw-Cutting  in  a  Lathe  .  .  .    30S 
Of  Steam  and  the  Steam  Ex* 

gine 308 

Velocity  of  Projectiles,  &c  .  ...    313 
Steam, "acting  expansively  .  .   .  .    313 
Of  the  Eccentric  in  a  Steam  En- 
gine   314 

Of  Continuous  Circular  Mo- 
tion  314 

To  find  the  number  of  Revolu- 
tions made  by  the  last,  to  one 
revolution  of  the  first,  in  a  train 
of  Wheels  and  Pinions    ....    315 
The  distance  from  Centre  to  Cen- 
tre of  two  Wheels  to  work  in 
contact  given,  and  the  ratio  of 
Velocity  between  them,  to  find 
their  Requisite  Diameters  .  .   .    317 
To  find  the  Velocity  of  a  Belt   .  .   317 
To  find  the  Draft  on  a  Machine    .   317 
To  find  the    devolutions  of  the 

Throstle  Spindle 318 

To  find  the  Twist  given  to  the 

Yarn  by  the  Throstle 318 

Tkktii  OF  Wheels,  &c 318 

To  construct  a  Tooth,  &c 319 

To  find  the   Horse-Power   of  a 

Tooth 319 

Journals  of  Shafts 880 

Hydrostatics 820 

hydraulics 322 

Water-  Win.  i:i.  I 888 

To  find  UM  Tower  of  a  Stream  .   .    881 
To  c.n-tni.-i  .i  Water-Wheel  to  a 

D  Power  and  Pall 325 

Dynamics 320 

Hydrostatic  Press 320 


SECTION    VI. 

IUO0,  (MB  l'ROB- 

•   <    I    II  IN... 

Remarksanp  Di.i  iNMiuNs  .  .  ,887 


To  construct  a  Pattern  for  the 
Lateral  Portion  of  a  vessel  In  the 
form  of  a  Frustum  of  a  Cone  of 
given  diameters  and  depth   .   .   .  329 

To  (.instinct  a  Pattern  for  the 
Body  of  a  vessel  In  the  form  of  a 
Frustum  of  •  Cone  of  given  di- 
mensions, without  plotting  the 
dimensions 332 

To  construct  a  Pattern  for  the 
Lateral  Portion  Of  a  Flaring  Ves- 
sel of  given  symmetry  of  outline 
and  given  capacity   ." 333 

TABLE  OF  Relative  Propor- 
tions, Chords,  &e 333 

The  special  tabular  figure,  the  di- 
ameter of  one  end,  and  the  Cubic 

Capacity  of  the  vessel  being 
given,  to  find  the  diameter  of 
the  other  end 330 

To  construct  a  Pat  tern  for  the  body 
of  a  Haring  Vessel  of  gives 
tabular  outline,  and  given  dimen- 
sions, without  plotting  the  di- 
mensions . 338 

The  Capacity  in  gallons  of  a  vessel 

.  In  the  form  of  a  Frustum  of  a 

Cone  being  given,  and  any  two 

of  its    dimensions,  to  find  the 

other  dimension 340 

To  construct  Patterns  for  flaring 
oval  vessels  of  different  eccentri- 
cities and  given  dimensions,  Nos. 
1,2,3 S|8 

To  describe  the  bases  for  Nos.  1,2, 8, 843 

OF  CYLINDRICAL  Ei.nows  .    .   .    .348 

To  construct  a  Pattern  for  a  Right- 
angled  Cylindrical  Elbow  .  .   .   ,810 

To  construct  Oblique-angled  El- 
bows   352 

To  construct  Right-angled  Elliptic 
Elbows 353 

To  construct  Oblique-angled  Ellip- 
tic Elbows 353 

To  construct  Right  Semi-liyperho- 
las  by  intersecting  lines    '.    ,840,863 

To  construct  the  Quadranl  of  a  Cir- 
cle by  intersecting  lines     .    .    . 

To  construe!  the  Quadrant  of  a 

J  riven  Ellipse  by  intersecting 
(net 354 

To  construct  the  Quadrant  of  a  Qy- 

cloidal  Ellipse  bv  intersecting 
lines ". 

TO  describe  an  Ellipse  of  given  di- 
mension- b\  means  of  t  w  u  I',. 
a  Pencil,  au  ■  String 

To  find  the  length  of  the  circum- 
ference of  a  given  Ellipse  .  . 

To  construct  a  Semi-parabola  by 
Interaafltlng  Hnti 355 


CONTENTS. 


PAOK 

Ovals,  to  describe    .  178,  343,  MS,  :t47 

Of  Circular  Kmiows 355 

TABUS  applicable  to  Circular  El- 
bows     350 

To  construct  a  Right-angled  Circu- 
lar  BlbOW  of  3,  4,  5,  6,  7,  or  8 

pieces,  &c 355 

To  construct  a  Collar  for  a  Cylin- 
drical I'ipc  of  the  same-  diameter 

as  the  receiving  pipe 359 

To  construct  a  Cylindrical  Collar 
of  a  given  Diameter  to  lit  a  Ue- 
ecivinu-pipe  of  a  greater  given 

Diameter 300 

To  construct  a  Cylindrical  Collar 
to  lit  an  Elliptic-cylinder  at  ei- 
ther right  section  of  the  El- 
lipse     301 

To  construct  a  Cylindrical  Collar 
of  a  given  Diameter,  to  fit  a  Cyl- 
inder of  the  same  Diameter,  at 
any  given  Angle  to  the  side  of 

the  Cylinder 301 

To  construct  a  Cylindrical  Collar, 
or  Spout,  of  a  given  Diameter, 
to  fit  a  Cylinder  of  a  greater  giv- 
en Diameter,  at  a  given  Angle 
to  the  side  of  the  Cylinder   .  .  .302 
Of  Spouts  for  Vessels    ....  303 
Of  Pitched  or  Bevelled  Covers   .  .  304 
To  construct  a  Bevelled  Circular 
Cover  of  a  given  Rise  and  giv- 
en Diameter 304 


PAGE 

To  construct  a  Pattern  for  a  Bev- 
elled Elliptical  Cover  of  a  given 
Rise  to  fit  an  Elliptic  Boiler  of 
given  Diameters 305 

To  construct  a  Bevelled  Cover  of  a 
given  Rise,  to  fit  a  False-Oval 
Boiler  of  given  length  and  width  305 

Of  Can-tops 306 

To  construct  a  Can-top  of  a  given 
Depth  and  given  Diameters  .  .   306 

To  construct  a  Can-top  of  a  given 
Pitch,  and  given  Diameters    .   .    867 

OF  Lips  for  Mkasures    .  .  .  .  308 

To  construct  a  Lip  for  a  Measure, 
the  Diameter  of  the  Top  of  the 
Measure  being  given 309 

OfShBBTPAJTI 309 

To  cut  the  Corners  for  a  Perpen- 
dicular-sided Sheet  Pan   ....   370 

To  cut  the  Corners  for  an  Oblique- 
sided  Sheet  Pan 370 

To  construct  a  Heart,  or  Heart- 
shaped  Cake-Cutter 370 

To  construct  a  Mouth-piece  for  a 
Speaking-Tube 370 

To  construct  a  Pattern  for  the 
Body  of  a  Circular  -  bottomed 
Flaring  Coal-Hod,  all  the  curves 
-to  be  arcs  of  circles 371 

Solders,  Alloys,  and  Compo- 
sitions     373 


DEFINITIONS 

OF  THE  SIGNS  USED  IN  THE  FOLLOWING  WORK. 


es  Equal  to.     The  sign  of  equality ;  as  16  oz.  =  1  lb. 

-f-  Plus,  or  More.     The  sign  of  addition  ;  as  8  -}-  12  =  20. 

—  Minus,  or  Less.     The  sign  of  subtraction  ;  as  12  —  8  =  4. 

X  Multiplied  by.     The  sign  of  multiplication  ;  as  12  X  8  =  96. 

-J-  Divided  by.     The  sign  of  division  ;  as  12  -r-  4  =  3. 

*r  Difference  between  the  given  numbers  or  quantities;  thus,  12  s>  8,  or 
8  is*  12,  shows  that  the  less  number  is  to  be  subtracted  from  the 
greater,  and  the  difference,  or  remainder,  only, .is  to  be  used  ;  so, 
too,  height  j-  breadth,  shows  that  the  difference  between  the  height 
and  breadth  is  to  be  taken; 

:  ::  :  Proportion;  as  2  :  4  ::  3  :  6  ;  that  is,  as  2  is  to  4,  so  is  3  to  6. 

V  Sign  of  the  square  root ;  prefixed  to  any  number  indicates  that  the 
square  root  of  that  number  is  to  be  taken,  or  employed ;  as 
V64  =  8. 

^/  Sign  of  the  cube  root ;  and  indicates  that  the  cube  root  of  the  num- 
ber to  which  it  is  prefixed  is  to  be  employed,  instead  of  the  num- 
ber itself;  as  ^64  =  4. 

8  To  be  squared,  or  the  square  of;  shows  that  the  square  of  the  number 
to  which  it  is  affixed  is  the  quantity  to  be  employed ;  as  122  -r- 
6  =  24  ;  that  is,  that  the  square  of  12,  or  144  -r-  6  =  24. 

8  Indicates  that  the  cube  of  the  number  to  which  it  is  subjoined  is  to 
to  be  used ;  as  43  =  64. 

•  Decimal  point,  or  separatrix.    See  Decimal  Fractions. 

Vinculum.    Signifies  that   the  two  or  more  quantities  over 

which  it  is  drawn,  are  to  be  taken  collectively,  or  as  forming 
one  quantity ;  thus,  4  +  6  X  4  =  40  ;  whereas,  without  the 
vinculum,  4  -f  6  X  4  =  28  ;  also,  12  —  2X3+4  =  2  ;  and 
V52ZZ32=S4.  So,  also,  V(52  — 32)=-4,and(44-6)X4 
=  40. 

4_2  (  half  of  42  or  ) 

2    (  half  of  the  square  of  4  )  ~ 

(42  \2 
-    J  (the  square  of  half  the  square  of  4)  =  64. 

i^2  or  4 (6)2  (half  the  square  of  b.) 
(hh)-  (the  square  of  half  b  ) 
(2b)2  (the  square  of  twice  b.) 


SECTION  I. 
MONEYS,  WEIGHTS  AND  MEASURES, 

OP    THE   UNITED  STATES  ;— THEIR  DENOMINATIONS,  VALUES, 
COMPARATIVE  VALUES,  MAGNITUDES,  Ac. 

MONEYS  OF  ACCOUNT  OF  THE  UNITED  STATES. 

These  are  the  mill,  the  cent,  the  dime,  and  the  dollar. 
10  mills  =ss  1  cent,  10  cents  =  1  dime,  10  dimes  =»  1  dollar. 

The  dollar  is  the  unit  or  ultimate  money  of  account  of  the  United 
States,  or  of  what  is  sometimes  called  Federal  money. 

In  practice,  the  dime,  as  a  denomination  of  value,  is  rejected. 
Thus, 

10  mills  =  1  cent,  and  100  cents  =  1  dollar. 

This  mark,  $,  is  equivalent  to  the  word  dollar,  or  dollars,  in  this 
money. 

COINS  OF  THE  UNITED   STATES. 

Until  June,  1834,  the  government  of  the  United  States  estimated 
gold  in  comparison  with  silver  as  15  to  1,  and  in  comparison  with 
copper  as  850  to  1. 

From  June,  1834,  until  February,  1853,  the  same  government 
estimated  gold  in  comparison  with  silver  as  16  to  1,  and  in  com- 
parison with  copper  as  720  to  1. 

For  all  time  since  February,  1853,  this  government  has  estimated 
gold  in  comparison  with  silver  as  14  £  to  1,  and  in  comparison  with 
copper  as  720  to  1. 

The  standard  for  mint  gold  with  this  government  until  1834, 
was  11  parts  pure  gold  and  1  part  alloy,  the  alloy  to  consist  of 
silver  and  copper  mixed,  not  exceeding  one  half  copper. 

The  gold  coins,  therefore,  struck  at  the  United  States  mint  prior 
to  1834,  are  22  carats  fine. 
2 


14  CURRENCY   OP  THE  UNITED  STATES. 

In  what,  until  1834,  constituted  a  dollar  of  gold  coin  of  United 
States  mintage,  there  were  put  24.75  grains  of  pure  gold ;  and  27 
grains  of  the  standard  mint  gold  of  that  day  were  at  that  time  worth 
$1.  Twenty-seven  grains  of  that  gold,  or  gold  of  that  standard, 
are  now,  by  the  present  government  standard  of  valuation,  worth 
$1.0652. 

The  standard  for  mint  silver  with  this  government  until  1834, 
was  1485  parts  pure  silver  and  179  parts  pure  copper,  =  8^fe 
parts  pure  silver    and  1  part  pure  copper. 

The  silver  coins,  therefore,  struck  at  the  United  States  mint  prior 
to  1834,  are  lOf  f  f  ounces  fine. 

In  that  which,  until  1834,  constituted  a  dollar  of  silver  coin  of 
this  government's  mintage,  there  were  put  37l£  grains  of  pure 
silver ;  and  416  grains  of  the  standard  mint  silver  of  that  day  were 
at  that  time  of  the  value  of  $1.  Four  hundred  and  sixteen  grains 
of  that  silver,  or  silver  of  that  standard,  are  now,  by  the  present 
government  standard  of  valuation,  worth  $1.0744. 

The  cent,  until  1834,  was  of  pure  copper,  and  weighed  208 
grains;  since  1834,  pure  copper,  weight  168  grains. 

The  standard  for  mint  gold  with  this  government  is  now,  and  for 
all  time  since  June,  1834,  has  been,  9  parts  pure  gold  and  one  part 
alloy,  the  alloy  to  consist  of  silver  and  copper  mixed,  not  exceeding 
one  half  silver. 

The  gold  coins,  therefore,  struck  at  the  United  States  mint  and 
dated  subsequent  to  1834,  are  21f  carats  fine. 

The  standard  weight  for  these  coins  is  25*  grains  to  the  dollar ;  and 
in  every  25f  grains  of  these  coins  there  are  23-j^ny  grains  of  pure 
gold. 

The  standard  for  mint  silver  with  this  government  is  now,  and 
for  all  time  since  June,  1834,  has  been,  9  parts  pure  silver  and  1 
part  pure  copper. 

The  silver  coins,  therefore,  struck  at  the  United  States  mint  and 
dated  subsequent  to  1834,  are  lOf  ounces  fine. 

In  what,  from  June,  1834,  until  February,  1853,  constituted  a 
dollar  of  silver  coin  of  this  government's  mintage,  there  were  put 
371$  grains  of  pure  silver ;  and  412£  grains  of  the  standard  mint 
silver  of  that  day  (the  present  standard)  were  worth,  from  June, 
1834,  until  February,  1853,  $1.  Four  hundred  twelve  and  one 
half  grains  of  thin  standard  of  silver  are  now  worth,  by  the  present 
standard  of  valuation,  $1.0742. 

The  standard  weight  for  silver  coins  with  this  government  at 
present  is  384  grains  to  the  dollar. 

The  new  cent,  established  by  the  Congress  of  1856,  is  7  parts 
copper  and  1  part  nickel,  and  its  legal  weight  is  72  grain*. 

The  foregoing  is  not  applicable  to  the  three-cent  pieces  of  United 


CURRENCY   OF   THE   UNITED   STATES. 


15 


States  mintage.  These  pieces  were  ordered  by  the  Congress  of  1850- 
1851,  and  an  especial  standard  of  purity  was  assigned  them, 
viz.,  three  parts  silver  and  one  part  copper  ;  their  weigiit  was  fixed 
at  12§  grains  each,  and  their  current  value  at  three  cents  each. 
The  law  of  1853,  regulating  the  currency,  does  not  apply  to  these. 
They  are  now,  as  in  1851,  legally  the  same.  These  pieces  are  worth, 
even  now,  less  than  their  nominal  values,  compared  with  the  present 
standard  of  purity  and  weight  for  other  United  States  coins.  They 
are  worth,  by  this  comparison,  2.863  cents  each. 

In  the  preceding  calculations,  the  alloy  for  gold,  in  each  instance, 
was  taken  to  consist  of  equal  parts  of  silver  and  copper.  The  law, 
until  1834,  provided  that  it  should  consist  of  '  silver  and  copper 
mixed,  not  exceeding  one  half  copper;'  and  the  present  law  pro- 
vides that  it  shall  consist  of  '  silver  and  copper  mixed,  not  exceed- 
ing one  half  silver.' 

The  metals  used  as  alloys  were  taken  at  their  values  as  money. 
Federal  money  was  established  by  the  Congress  of  the  United 
States,  in  1786. 
Boston,  June,  1866. 

GOLD,  —  PURE. 

24  carats  fine  =  Pure  Gold. 

1      grain       =  $0.0429. 

23.30859  «        —  $1.00. 

1  dwt.  =  $1.02966. 

1  ounce  =$20.5932. 

MINT   GOLD.  —  U.    S. 
Alloy  half  each,  silver  and  cojrper. 
Nine  parts  pure  gold  and  one  part  alloy  ;  or, 

21|  carats  fine  =  Standard  Coin. 
1    grain  =$0  03876. 

254    "  =$1.00. 

1    dwt.  =$0.93023. 

1    ounce  =$18.60465. 

GOLD   COINS. —  U.    S. 


Double  Eagle, 

Eagle,    -        -        -        - 

Half  Eagle,    - 

Quarter  Eagle, 

Gold  Dollar,  - 

Triple  Gold  Dollar, 

Eagle,  prior  to  1834,  ($10£,) 

Half  do., "    "    »      ($5*,) 


Weight  In 

Standard 

Grains. 

Value. 

516 

$20.00 

258 

10.00 

129 

5.00 

64£ 

2.50 

25| 

1.00 

77| 

3.00 

270 

10.64 

135 

5.32 

16 


CURRENCY   OF   THE   UNITED   STATES. 


Private  and  Uncurrent. 

Wedffal  iu 

Grains. 

Sales. 

A.  Bechtler,  N.  0.,  $5  piece, 

$4.75 

m            M         2£  "     - 

- 

- 

2.37 

(<            <(         1     '*     - 

. 

. 

.93 

T.  Reed,  Georgia,       5    "     - 

- 

- 

4.75 

K                (4                      2^    "       - 

- 

. 

2.37 

M                 ((                        J        (( 

. 

. 

.93 

Moffat,  California,      5    "     - 

- 

- 

129 

5.00 

SILVER, —  PURE. 

12  ounces  fine   =  Pure  Silver. 
1  dwt.  =  $0.06928. 

3463|  grains  =  $i. 
1  ounce  =  $1.3857. 

MINT  SILVER. —  U.  S. 

Alloy,  all  copper. 

Nine  parts  pure  silver  and  one  part  alloy ;  or, 
10  oz.  16  dwts.  fine  =  Standard  Coin. 

1  dwt.  —  $0,062. 

384  grains  —  $1.00. 

1  ounce  =  $1.23958. 

SILVER   COINS. —  U.    S. 


Dollar, 

Half  Dollar, 

Quarter  Dollar,    - 

Dime,  .--..- 

Half  Dime, 

Three-Cent  Piece,  £  silver  and  £  copper, 


Weight  in 

Cniins. 


384 
192 

96 

38f 

19* 
12f 


Standard 
Value. 


$1.00 
.50 
.25 
.10 
.05 
.03 


The  copper  coins  of  the  United  States  are  the  cent  and  half  cent  ; 
thev  are  of  pure  copper.  The  weight  of  the  former  is  168  grains, 
and  that  of  the  latter,  84  grains. 

Notk.  — The  silver  coins  of  the  United  States,  issued  Bince  February,  1863,  are  not 
legal  tender  in  the  United  States  in  sums  exceeding./?i;e  dollars. 


CURRENCY   OF   TIIE   UNITED   STATES. 


17 


TABLE, 

Exhibiting  the  standard  weight  and  present  par  value  of  the  silver  coins 
of  the  United  States,  of  dales  subsequent  to  1834,  and  prior  to  1853. 


Weight  In 

Present 

Grains. 

par  value. 

Dollar,     - 

4124 

$1.0742 

Half  Dollar,      -         . 

206j 

.5371 

Quarter  Dollar,  - 

103* 

.2685 

Dime,       -         -         -         -         - 

41* 

.1074 

Half  Dime,        - 

m 

.0537 

Three-Cent  Piece,       - 

12| 

.03 

CURRENCIES   OF   THE   DIFFERENT   STATES   OF   THE   UNION. 

4  Farthings  =  1  Penny,  12  Pence  =  1  Shilling,  20  Shillings  =» 
1  Pound. 

In  Massachusetts,  Connecticut,  Rhode  Island,  New  Hampshire, 
Vermont,  Maine,  Kentucky,  Indiana,  Illinois,  Missouri,  Virginia, 
Tennessee,  Mississippi,  Texas  and  Florida,  6  shillings  =  1  dollar  ; 
$!  =  •&£• 

In  New  York,  Ohio  and  Michigan,  8  shillings  =  1  dollar  ;  $1  =a 

In  New  Jersey,  Pennsylvania,  Delaware  and  Maryland,  7  shil- 
lings and  6  pence  =*  1  dollar ;  1  dollar  =  %  £. 

In  North  Carolina,  10  shillings  =*  1  dollar ;  $1  =  £  £. 

In  South  Carolina  and  Georgia,  4  shillings  and  8  pence  »■  1  dol- 
lar ;  $1  =  ,&  £. 

Note. —  These  currencies,  so  called,  are  nominal  at  present  in  a  great  measure.  The 
denominations  serve  in  the  different  States  a3  verbal  expressions  of  value.  But  they  are 
neither  the  names  of  the  moneys  of  account  in  any  of  the  States,  nor  are  they  the  national 
names  of  any  of  the  real  moneys  in  circulation.  All  values  in  money  in  the  United  States 
are  legally  expressed  in  dollars,  cents,  and  mills. 

2* 


18  METRICAL  SYSTEM   OF   WEIGHTS   AND   MEASURES. 


THE  METRICAL   SYSTEM   OF  WEIGHTS  AND  MEAS- 
URES. 

In  this  system,  the  Metre  is  the  basis,  and  is  one  forty-millionth 
of  the  polar  circumference  of  the  earth. 

The  Metre  is  the  •principal  unit  measure  of  length;  the  Are 
of  surface;  the  Stere  of  solidity;  the  Litre  of  capacity;  and  the 
Gram  of  weight. 

The  gram  is  the  weight,  in  a  vacuum,  of  one  cubic  centimetre 
of  pure  water  at  its  maximum  density. 

The  Metre,  almost  exactly  .     ==     39.3685     U.  S.  inches. 

The  Are  (100  square  metres)   z=       3.95337      "      square  rods. 

The  Stere  (a  cubic  metre)  .     =z     35.31042      "     cubic  feet. 

t\     tu      s       ..    .    .      .   v        (61.0164        "         "      inches. 
The  Litre  (a  cubic  deC1metre)=:j    lMm      u     wine  quarts. 

The  Gram  .     =      15.44242     "     grains. 

The  divisions  by  10,  100,  1,000,  of  each  of  these  units,  are  ex- 
pressed by  the  same  prefixes,  viz.,  deci,  centi,  milli;  and  the  multi- 
ples by  10,  100,  1,000,  10,000,  of  each,  by  deca,  hecto,  kilo,  na/ria. 
The  former  series  were  derived  from  the  Latin  language,  the  latter 
from  the  Greek. 

To  illustrate  with  the  metre :  — 

10  millimetres  =i  1  centimetre,  10  centimetres  =:  1  decimetre,  10 
decimetres  =1  Metre,  10  Metres  =  1  decametre,  10  decame- 
tres =  1  /hectometre,  10  hectometres  =  1  Mometre,  10  kilometres 
=.  1  m^nametre. 

In  commerce,  the  ordinary  weight  is  the  kilogram,  and  100  kilo- 
grams (usually  called  kilos)  r=z  1  quintal;  10  quintals  =  1  millier, 
or  tonneau.  The  kilogram  =  15,442.42  -f-  7000  =  2.20606  avoir- 
dupois pounds. 

In  practice,  the  terms  milliare,  declare,  decare,  kiloare,  and  myri- 
are  are  usually  dropped,  and 

100  centare  =z  1  are;  100  ares  =  1  hectare. 

Also  the  terms  millistere,  hectostere,  kilostere,  and  myriastere,  are 
usually  rejected,  and  100  centisteres  z=  1  decistere;  10  decisteres 
■=.  1  stere  ;  10  stores  =.  1  decastere  zr:  353.1042  cubic  feet. 
1  centiare  (square  metre)     =    1.19589413  square  yards. 
1  kilometre      .         .         .    =    0.62135  statute  miles. 
1  hectare         .         .         .    E=    2.471  =  U.  S.  acres. 
1  kilolitre         .         .         .     =    1  stere  =  61,016.403233  cubic  in. 
A  hectolitre  =  26.41403  wine  gallons  =l  2.83741  Winchester  bush. 

Notk.  —  The  system  is  the  one  recommended  by  tbf  Statistical  OOBfW 
1865  aa  a  general  system  of  weights  ami  measure*  to  be  adopted  by  all  nations. 


fOREIGN    GOLD   COINS. 


19 


FOREIGN  GOLD   COINS. 


Note. —  The  coins  of  any  country,  both  gold  and  silver,  circulating  as  for- 
eign in  any  other,  particularly  those  of  the  smaller  denominations,  are  usually 
keld  at  an  estimate  below  their  standard  par  value,  compared  with  the  money 
standard  of  the  country  in  which  they  circulate  as  foreign.  Many  of  them, 
more  particularly  the  silver,  having  circulation  in  the  United  States,  are  much 
worn  and  otherwise  depreciated.  In  some  instances,  owing  to  frequent  changes 
made  both  with  regard  to  weight  and  purity,  certain  of  them,  having  the  same 
name  and  general  appearance,  boar  a  premium  at  home;  others,  a  discount. 
Others,  again,  can  hardly  be  said  to  have  a  definable  value  anywhere.  The  par 
value  of  the  old  pistole  of  Geneva,  for  instance,  weighing  103i  grains,  is  $3,985, 
while  that  of  the  new,  weighing  873  grains,  would,  at  the  same  degree  of 
purity,  be  worth  but  $3,386  ;  whereas,  owing  to  its  higher  standard  of  fine- 
ness, its  par  value  is  $3,443.  The  ducat  of  Austria,  coined  in  1831,  weighs 
53A  grains,  —  its  purity  is  23.64,  and  its  par  value  $2,269;  while  the  half 
sovereign,  closely  resembling  the  ducat,  coined  in  1835,  and  weighing  87  grains, 
has  a  purity  only  of  21.64,  and  a  par  value,  consequently,  of  but  $3,378.  The 
circulating  value  of  the  ducat  in  the  United  States,  in  general,  is  $2.20,  and 
that  of, the  haif  sovereign  of  Austria,  $3.25. 


Standard 

Standard 

Par  value 

Circulating 

Par  val- 

ARGENTINE   REPUBLIC. 

of 
parity  in 

weight 

in 
Federal 

value  in 
Federal 

ue  per 
grain. 

Doubloon  to  1832, 

carats. 

grains. 

money. 

■money. 

ct*. 

19.56 

418 

$14,671 

$ 

3.50 

to     " 

20.83 

415 

15.512 

3.73 

AUSTRIA. 

Sovereign,  half  in  propor- 

tion, to  1785, 

22.00 

170 

6.711 

6.50 

3.94 

Sovereign,  half  in  propor- 

tion, since  1785, 

21.64 

174 

6.756 

6.50 

3.88 

Ducat,  double  in  propor- 

tion, 

23.64 

53* 

2.269 

2.20 

4.24 

BELGIUM. 

Sovereign,  half  in  pro., 

22.00 

170 

6.711 

3.94 
— - — 1 

20 


FOBE1GK  GOLD  COINS, 


Standard 
of 

Standard 
weight 

Par  rsltw 
in 

Circulating 
rahie  in 

Par  »aT| 
ue  per   j 

parity  in 

in 

Federal 

Federal 

grain. 

Twenty  Franc,  more  in  pro. 

carats. 

gTaini. 

money. 

money. 

eta. 

21.50 

99£ 

$3,840 

$3.83 

3.85 

Ducat, 

2.20 

Bolivia,  Colombia,  Chili, 

Ecuador,    Peru,    New 

Grenada,   and   Mexico. 

Received  by  U.  S.  Gov- 

ernment, —  those  of  not 

less    than   20.86   carats 

fine,  at  89-j^  cts.  per  dwt. 

Doubloon,  (8  E) 

20.86 

417 

15.620 

15.60 

3.74 

Half  do. 

t< 

208£ 

7.810 

7.50 

M 

Quarter  do. 

II 

1044 

3.905 

3.75 

M 

Eighth  do. 

M 

52 

1.952 

1.75 

II 

Sixteenth  do. 

M 

26 

.976 

.90 

(( 

Pistole,  half  in  pro., 

3.75 

BRAZIL. 

Received    by  the  U.    S. 

Government,  —  those  of 

not  less  than   22  carats 

fine,  at  94^  cts.  per  dwt. 

Dobraon, 

22.00 

828 

32.719 

32.00 

3.95 

Dobra, 

M 

438 

17.306 

17.00 

M 

Joannes,  {standard  variable) 

M 

432 

17.064 

Sl3to$17 

II 

Half  do.       do.        do. 

It 

216 

8.532 

$6  to  8.50 

(( 

Moidore,  (BBBB)  half  in 

pro.,    {standard  variable) 

21.79 

165 

6.451 

6.00 

3.90 

Crusado,      do.          do. 

M 

16* 

.635 

N 

DENMiP*. 

Christian  d'or 

21.74 

103 

4.018 

3.90 

Ducat,  species, 

23.48 

53£ 

2.254 

2.20 

4.21 

11       current, 

21.03 

48 

1.811 

3.77 

FRANCE. 

{Alloy  mostly  silver.) 

Rec'd   by  U.   S.  Govern- 

ment, —  those  the  purity 

of  which  is  not  less  than 

21^  carats  fine,  at  93^ 

cts.  per  dwt. 

Chr.  d'or,  double  in  pro., 

21.60 

101 

3.914 

3.90 

3.87 

FOREIGN   GOLD   COINS. 


21 


Standard 

Btaad  utJ 

Var  value 

Circulating 

Par  val- 

of 

weight 

in 

value  in 

ue  per 

purity  in 

in 

Federal 

federal 

grain. 

Franc  d'or,  double  in  pro., 

emus. 

grains. 

money. 

money. 

ct». 

21.60 

101 

$3,914 

$3.90 

3.87 

Louis  d'or,      "       "    " 

to  1786, 

21.49 

125£ 

4.840 

3.85 

Louis  d'or,  double  in  pro., 

since  1786, 

21.68 

118 

4.573 

4.50 

3.87 

Napoleon  (20  F.)  double  &c. 

21.60 

994 

3.856 

3.83 

(4 

GERMANY. 

BADEN. 

Zehn  Gulden,  5  in  pro.. 

21.60 

1054 

4.088 

4.00 

3.87 

BAVARIA. 

Carolin, 

18.49 

149£ 

4.952 

3.32 

Ducat,  double  in  pro., 

23.58 

53| 

2.275 

2.20 

4.23 

Maximilian, 

18.49 

100 

3.317 

3.31 

BRUNSWICK. 

Ducat, 

23.22 

534 

2.220 

4.16 

Pistole,  double  in  pro., 

21.60 

117^ 

4.548 

3.87 

Ten  Thaler,  5  in  pro.,  to 

1813, 

21.55 

202 

7.811 

7.80 

3.86 

Ten  Thaler,  less  in  pro., 

since  1813, 

21.50 

204 

7.873 

7.80 

3.85 

HANOVER. 

Ducat,          double  in  pro., 

23.83 

534 

2.287 

2.20 

4.27 

George  d'or,    "       "    " 

21.67 

1024 

3.987 

3.88 

Zehn  Thaler,  5       "    " 

21.36 

2044 

7.838 

7.80 

3.83 

HESSE. 

Ten  Thaler,  5  in  pro.,  to 

1785, 

21.36 

202 

7.742 

u 

Ten  Thaler,  5  in  pro.,  since 

1785, 

21.41 

203 

7.799 

3.84 

SAXONY. 

Ducat, 

23.49 

534 

2.256 

2.20 

4.21 

Augus 

tus  d'or,  double  in 

pro. 

,  since  1784. 

WURTEMBURG. 

21.55 

1024 

3.964 

3.86 

Carolii 

k. 

18.51 

1474 

4.899 

3.32 

Ducat, 

23.28 

534 

2.235 

4.17 

22 


FOREIGN  GOLD  COINS. 


Standard 

Standard 

Par  ralue 

Circulating 

Par  ral- 

of 

weight 

in 

Talue  in 

ue  per 

purity  in 

in 

Federal 

Federal 

grain. 

GREAT    BRITAIN. 

carats. 

grain*. 

money. 

money. 

CtM. 

(Alloy,  since  1826,  all  copper.) 

Rec'd  by  U.   S.  Govem- 

ment,  —  those  of  22  ca- 

rats  fine,    set  94y^y    cts. 

per  dwt. 

Guinea,  half   h*  pro.,   to 

;     1785, 

22.00 

127 

$5,016 

3.95 

Guinea,  half  in  pro.,  since 

1785, 

« 

1294 

5.111 

$5.00 

<( 

Sovereign,  half  in  pro., 

M 

123£ 

4.866 

4.83 

<( 

iFive  do. 

U 

616| 

24.332 

24.20 

tt 

Sovereign,  (dragon)  half 

in  pro., 

It 

1224 

4.838 

4.80 

a 

Double  Sovereign  (dragon} 

u 

246 

9.717 

9.67 

u 

GREECE. 

Twenty  Drachm,  more  hi 

pro., 

21.60 

89 

3.449 

3.30 

3.87 

HOLLAND. 

• 

Ducat, 

23.58 

53£ 

2.263 

2.20 

4.23 

Ryder, 

22.00 

153 

6.043 

3.95 

Double  do. 

M 

309 

12.205 

(i 

Ten  Gulden,  5  in  pro., 

21.60 

104 

4.025 

4.00 

3.87 

INDIA. 

\ 

Pagoda,  star, 

19.00 

52| 

1.798 

3.40 

Mohur,  (E.  I.  Co.)  1835. 

22.00 

180 

7.106 

6.75 

3.95 

Half  Sovereign,  do. 

2.41 

BOMBAY. 

Rupee, 

22.09 

179 

7.095 

3.96 

MADRAS. 

Rupee, 

22.00 

180 

7.106 

3.95 

ITALY. 

Eturia,  Ruspone, 

23.97 

1611 

6.935 

4.30 

Genoa,    Sequin, 

23.86 

53£ 

2.291 

4.28 

Milan,    Pistole, 

21.76 

974 

3.807 

3.90 

"         Sequin, 

23.76 

534 

2.281 

4.26 

FOREIGN  GOLD  COINS. 


28 


Standard 

Stnndnrd 

Par  value 

Circulating 

Par  val- 

of 

weight 

in 

value  in 

ue  per 

purity  in 

in 

Federal 

Federal 

grain. 

Milan,  Twenty  Lire,  more 

carats. 

."r.iinv 

money. 

money. 

ett. 

in  proportion, 

21.58 

99£ 

$3,853 

$3.83 

3.86 

Naples,  Ducat,  multiples 

in  pro., 

21.43 

22£ 

.865 

3.84 

Naples,  Oncetta, 

23.88 

58 

2.485 

4.28 

Pauma,  Doppia,  to  1786, 

21.24 

110 

4.192 

3.81 

"       Pistole,  since  1796, 

20.95 

no 

4.135 

3.75 

"       Twenty  Lire, 

21.62 

994 

3.860 

3.83 

3.87 

Piedmont,  Carlino,  half  in 

pro.,  since  1785, 

21.69 

702 

27.321 

3.89 

Piedmont,  Pistole,  half  in 

pro.,  since  1785, 

21.54 

140 

5.411 

3.86 

Piedmont,  Sequin,  half  in 

• 

pro.,  since  1785, 

23.64 

53i 

2.280 

4.23 

Piedmont,  Twenty   Lire, 

more  in  pro. , 

20.00 

991 

3.563 

3,50 

3.59 

Rome,  Ten  Scudi,  5  in  pro. 

21.60 

2674 

10.368 

3.87 

"      Sequin,  since  1760, 

23.90 

52i 

2.251 

4.28 

Sardinia,   Carlino,    £    in 

pro., 

21.31 

2474 

9.465 

3.82 

Tuscany,  Zeehino,  double 

in  pro., 

23.86 

531 

2.302 

4.30 

Venice,    Zeehino,  double 

in  pro., 

23.84 

54 

2.310 

MALTA. 

Sequin, 

23.70 

53£ 

2.275 

4.25 

Louis    d'or,    double    and 

demi  in  pro., 

20.25 

128 

4.651 

3.63 

NETHERLANDS. 

Dueat, 

23.52 

534 

2.257 

4.21 

Zehn  Gulden,  5  in  pro., 

21.55 

1031 

4.013 

4.00 

3.86 

POLAND. 

Ducat, 

23.58 

534 

2.264 

4.23 

PORTUGAL. 

Rec'd  by  the   U.  S.  Gov- 

ernment, —  those  the  pu- 

rity of  which  is  not  less 

than  22   carats  fine,  at 

94-j^  cts.  per  dwt. 

24 


FOREIGN   GOLD  COINS. 


Standard 

Standard 

Par  value 

Circulating 

Par  val- 

of 

weight 

in 

value  in 

ue  per 

purity  in 

in 

Federal 

Federal 

grain. 

Dobraon,  24,000  reis, 

carats. 

grains. 

money. 

money. 

ctt. 

22.00 

828 

$32,706 

$32.00 

3.95 

Dobra, 

<( 

438 

17.301 

17.00 

u 

Joannes,  {standard  variable) 

a 

432 

17.064 

813  to  31 7 

a 

Half  "           "           " 

(< 

216 

8.532 

86  to  8.50 

<i 

Moidore,  4000  reis,     M 

m  to  $4} 

Coroa,      5000    " 

" 

147£ 

5.83 

5.75 

«< 

Milrea, 

22.00 

191 

.780 

3.95 

PRUSSIA. 

Ducat, 

23.49 

53£ 

2.255 

2.20 

4.21 

Frederick  d'or,  double  in 

pro., 

21.60 

102£ 

3.973 

3.87 

RUSSIA. 

Ducat, 

23.64 

54 

2.291 

4.24 

Imperial,  (10  R.)  half  in 

pro.,  1801, 

23.55 

185$ 

7.828 

4.22 

Imperial,  (10  R.)  half  in 

pro.,  since  1818, 

22.00 

199 

7.856 

7.80 

3.95 

SICILY. 

Oncia,  double  in  pro., 

20.39 

68£ 

2.495 

3.64 

Twenty  Lire,  more  in  pro., 

21.60 

99* 

3.856 

3.83 

3.87 

SPAIN. 

Rec'd  by  U.  S.  Govern- 

ment, —  those  the  stand- 

ard purity  of  which  is 

not  less  than  20.86  ca- 

rats fine,  at   89 ^   cts. 

per  dwt. 

Doubloon  (8  S)  parts  in  pro. 
"       (8   E)    parts   as 

21.45 

416* 

16.031 

16.00 

3.84 

Bolivian,  &c. 

20.86 

417 

15.620 

15.00 

3.74 

Pistole,  to      1782, 

21.48 

103 

3.970 

3.85 

"       since    " 

20.93 

101 

3.906 

3.75 

Escudo,  to      1788, 

20.98 

52 

1.957 

1.76 

"       since   " 

20.42 

52 

1.905 

3.W. 

Coronilla  "    1800, 

20.29 

27 

.983 

B.64 

SWEDEN. 

Ducat, 

23.45 

53 

2.230 

4.20 

LONG   OR    LINEAR    MEASURE. 


25 


SWITZERLAND. 

Berne,   Ducat,  double  in 

pro., 
Berne,  Pistole, 
Geneva,  Pistole, 

"         {old) 
Zurich,  Ducat,  double  in 

pro., 

TURKEY. 
Misseir,  half  in  pro.  1820, 
Sequin  fonducli, 
Yeermeeblekblek, 


J  Standard 

Standard 

Par  value 

Circulating 

of 

weight 

in 

value  in 

purity  in 

in 

Federal 

Federal 

carats. 

grams. 

■money. 

money. 

23.53 

47 

$1,984 

21.62 

117£ 

4.558 

21.87 

87$ 

3.443 

21.51 

1034 

3.985 

23.50 

53* 

2.256 

15.88 

36£ 

1.040 

19.25 

53 

1.830 

22.88 

73| 

3.027 

Par<ral 

ue   per 


4.22 
3.88 
3.92 
3.85 

4.21 


2.84 
3.45 
4.10 


Note.  — The  Milled  Dollars,  or  Pesos  (silver)  of  Spain,  Mexico,  Peru,  Chili,  and  Cen- 
tral America,  and  the  re  stamped  of  Brazil,  weighing  not  less  than  415  grains,  and  of  10 
oz.  15  dwts.  fu*e,  are  received  by  the  United  States  Government  at  $1.00  each. 

The  Five  Franc  silver  pieces  of  France,  of  10  oz.  16  dwts.  fine,  and  weighing  384 
grains,  are  received  at  i)3  cents  each. 

The  standard  silver  coins  of  Great  Britain  are  11  oz.  2  dwts.  fine. 


LONG  OR  LINEAR  MEASURE.  —  U.  S. 

Standard.  —  A  brass  rod,  the  length  of  which,  at  62°  Fahrenheit, 
is  f  f  ;^§jf§  that  of  a  pendulum  beating  seconds  in  vacuo,  at  the  level 
of  the  sea,  at  the  latitude  of  London,  =  $f;T^#  at  32°  Fah.,  at  the 
gravitation  at  New  York,  —  the  Yard. 


6  points  =  1  line. 

12  lines  (72  points)  =  1  inch. 

12  inches  =  1  foot. 

3  feet  (36  inches)  =  1  yard. 


5£  yards  (16£ft.)  =  l  rod. 
40  rods  (220  yds.)  =  1  furlong. 
8  fur.  (5280  feet)  =  1  stat.  mile. 


2|  inches  =  1  nail. 

4  nails  (9  inches)    =  1  quarter 


SPECIAL,    FOR    CLOTH. 

4  quarters  (36  inches)    =  1  yard. 


7-j-9^  inches 
25  links 


SPECIAL,    FOR    LAND. 

=  1  link.  I  100  links  (66  feet)     =  1  chain. 

=  1  rod.   |  80  chains  (320  rods)  =  1  s.  mile. 

engineer's  chain. 

10  inches  =  1  link. 

120  links  (100  feet)     =  1  chain. 


SQUARE    OR    SUPERFICIAL   MEASURE. 


SHOEMAKER'S  MEASURE. 

No.  1  is  4 J  inches  in  length,  and  each  succeeding  number  is  an 
addition  of  J  of  an.  inch.    No.  1  man's  size  =  8£|  inches. 

MISCELLANEOUS. 

Fathom  ==  6  feet. 

Knot  =  471  feet. 


Hair's  breadth 
Digit 


Cable's  length     =  120  fathom*. 
Geometrical  pace=  4.4  feet. 


=  5^  inch. 
=  10  lines. 
Palm  =   3  inches. 

Hand  =    4     " 

SpaD  ==    9     «* 

12  particular  things  =  1  dozen. 

12  dozen  (144)  =  1  gross. 

12  gross  (1728)         =  1  great  gross. 

20  particular  things  =  1  score. 

24  sheets  of  paper     =  I  quire. 

20  quires  =  1  ream. 

SQUARE  OR  SUPERFICIAL  MEASURE. 
(Length  X  breadth.) 
144  square  inches  =  1  square  foot, 
yard. 


9 

304 

40 

4 


feet  =  1 

yards  =  1 

rods  =  1 

roods  =  1 


rod. 

rood. 

acre. 


SPECIAL,    FOR    LAND. 


srisuiAL,    run    l,as\u. 

62|^4  square  inches  =  1  square  link. 
"       links    =  1       " 


10000 
10 
Square  rod 

Rood 


chains  =  1 


chain, 
acre. 
272$  square  feet, 
yards 


Acre  ( 160  square  rods)  == 

Square  mile 

220  X  198  square  feet 

The  square  of  12.649  "     rods 

"         "     of   69.5701        "     yards 
"        "     of  208.710321    "     feet 


1210 

10890 

4840 

43560 

640 

102400 


feet. 

yards. 

feet. 

acres. 

rods. 


sq.      rods. 
>  =1  acre. 


CUBIC   OR    SOLID   MEASURE. 


27 


CIRCULAR  MEASURE. 


Minute,  or 
Geogra- 
phical m. 
(60") 

League 

Degree 


1.152  s.  miles. 
-^  6086  feet. 


-f 


=  360  degiees. 


=  <  24897  s.  ra. 


Great  Circle 
Equatorial  cir- 
cumference 
of  the  earth 
Equatorial  diam.=  7925 
Polar  diam.         =  7899 
Mean  radius        =  3955.92 


hi 


viz., 


=  3  miles. 

60  geo.  miles. 
69.158  s.  ms. 
Sign(-j^  zod.)=30  degrees. 

Note.  —  In  the  expressions,  square  feet  and  feet  square,  there  is  this  difference ; 
the  former  expresses  an  area  in  which  there  are  as  many  square  feet  as  the  number 
named,  and  the  latter  an  area  in  which  there  are  as  many  square  feet  as  the  square  of  tha 
number  named.    The  former  particularizes  no  form  of  area,  the  latter  asserts  a  squam 

CUBIC  OR   SOLID  MEASURE.  — U.  S. 
(Length  X  breadth  X  depth.) 

f  1.273  cylindrical  feet. 
>  __  J  2200  "         inches. 

$         )  3300  spherical  " 

(_6600  conical  " 

f  0.785398  cubic  feet. 
J  1357.2        "      inches. 
)  2592  spherical       " 
1 5 184  conical  " 

27  cubic  feet  =      1  cubic  yard. 

40         "      of  round  timber  =  1  ton. 
42  "      of  shipping    "   =  1  ton. 

50  "      of  hewn         "   =  1  ton. 

128  "  =  1  cord. 

Cubic  foot  of  pure  water,"! 

at  the  maximum  density  J  ,  cn.        .  .  . 

at  the  level  of  the  sea,  I  =  \  62^  avoirdupois  pounds. 
(39°.83,   barometer  30  <  1UUU 

inches) .  .  J  49.1 

=  }  785.4 


Cubic  foot, 
1728  cu.  inches 


Cylindrical  foot 
1728    "   inches 


h 


ounces. 


Cylindrical  foot 


Cubic  inch 


=  Jo. 

(25 


Cylindrical  inch  = 

Pound  = 

"     distilled  = 

Cubic  inch  u  = 

Pound  at  62°,  distilled  = 

Cubic  inch  at  62°,  "  = 
"         "    39°.83,  in  vacuo  = 


036 169" 

5787     " 
253.1829 
0.028415  avd. 
0.4546        " 
27.648  cubic  inches. 
27.7015    " 
252.6839  grains. 
27.7274  cub.  inches. 
252.458  grains. 
253.0864     " 


pounds. 

ounces. 

pounds. 

ounces. 

grains. 

pounds. 

ounces. 


Cubic  foot  of  salt  water  (sea)  weighs  64.3  pounds. 


28 


GENERAL   MEASURE    OP   WEIGHT. 


GENERAL    MEASURE  OF  WE 

AVOIRDUPOIS. 


a 


Standard.  —  The  pound  is  the 
weight,  taken  in  air,  of  27.7015 
cubic  inches  of  distilled  water  at 
its  maximum  density,  (39°.83  F., 
the  barometer  being  at  30  inches) 
=  27.7274  cubic  inches  of  distilled 
water  at  62°  =  7000  Troy  grains. 

27^£  grains  =  1  dram. 

16  drams  (437£  grs.)  =  1  ounce. 
16  ounces  (7000  grs.)=  1  pound. 

SPECIAL GROSS. 

28  pounds 

4  quartere  > 
112  pounds    5 
20cwt. 

SPECIAL DIAMOND. 

16  parts  =  1  grain  =  0.8  troy  gr. 
4  grs.   =1  carat  =  3,2    "    " 

Note. 


=        1  quarter. 
5  1  quintal. 


cwt. 
1  ton. 


SPECIAL TROT. 

(Exclusively  for  gold  and  sil- 
ver bullion,  precious  stones,  and 
gold,  silver  and  copper  coins,  and 
with  reference  to  their  monetary 
value  only.) 

24  grain&  =  1  penny w't. 

20dwts.  (480  grs.)=  1  ounce. 
12  oz.  (5760  grs.)  =»  1  pound. 

SPECIAL APOTHECARIES'. 

(Exclusively  for  compounding 
medicines,   for   recipes  and   pre- 
scriptions.) 
20  grains  =  1  scruple,  B>. 

3  scruples  =  1  dram,       5. 

8  drams(480  g.)=  1  ounce,     5. 
12  oz.  (5760  g.)  =  1  pound,    lb. 

1  lb.  avoir.  =  lj3^  lbs.  troy. 

1  lb.  troy  =    IH  lbs.  avoir. 

1  oz.  avoir.  =    \^§  oz.   troy. 

1  oz.  troy  =  ly^  oz.  avpir. 

The  comparative  vaTuc  of  diamonds  of  the  same  quality  is  as  the  square  of 
Iheir  respective  weights.  A  diamond  of  fair  quality,  weighing  1  carat  in  the  rough  state^ 
ib  estimated  worth  about  89-j-^j.  ;  and  it  will  require  one-of  twice  that  weight  to  make 
one  when  worked  down  equal  to  1  carat  in  weight.  Hence,  to  determine  the  value  of  a 
wrought  diamond  of  any  given  numt>er  of  carats :  —  Rule.  —  Double  the  weight  in  carata 
and  multiply  the  square  by  9.50.  Thus,  the  value  of  a  wrought  diamond,"  weighing  2 
carats,  is  2-f2=4  X  4  =  16  X  9^0=8152l 


LIQUID  MEASURE.  —  U.  S. 

The  "  Wine"  or  "  Winchester"  Gallon,  of  231  cubic  inches 
capacity,  is  the  Government  or  Customs  gallon  of  the  Unitqd  States 
for  all  liquids,  and  the  legal  gallon  of  each  state  in  which  no  law 
exists  fixing  a  state  or  statute  gallon  of  its  own.  It  contains  58372| 
grains  of  distilled  water  at  39°. 83,  the  barometer  being  at  30  inches. 

4  gills    =     1  pint,     2  pints    =     1  quart. 

4  quarts,  or  231  cubic  in.  ) $  I  gallon. 

t.  J-J8.2 


0.13368  cub.  ft.,  294.1176  cyl.  in. 


.355  av'd.  lbs.  pure  water. 


DRY  MEASURE.  29 

0.128  cubic  foot, 
in. 


w     -j      u         c  *u    x       fO.128  cubn 
Liquid  gallon  of  the  )         001104  « 

iTrf  1:7  York*r    8-01'    lbs.  pure  water 
281.62  cylindnc  in.  )       ^  ^^  b  ^  {a 

Barrel        «=        31£  gallons.  I  Puncheon        a*        84  galloi 
Tierce        *■        42        M  Pipe  or  Butt   «=      126      " 

Hogshead  =        63        "       |  Tun  =      252      " 


Imperial  gallon,  > { 

277.274  cub.  in.  {  ~  \ 


10  av'd  lbs.  distilled  water 
at62°F.,b.  36  in. 


Ale  gallon,  £  _  J  10£  av'd  lbs.  pure  water 
282  cub.  in.  J       |      at  39°.83,  b.  30  in. 


0.8331  Imperial  gallon, 
1  Wine  gallons {  0.8191  Ale  " 

16742  W.  bushel. 


(0.1 
1  Imperial  gallon        =        1.2        Wine  gallons. 


DRY  MEASURE.  —  U.  S. 

The  "  Winchester  Bushel,"  so  called,  of  21503^  cubic  inches 
capacity,  is  the  Government  bushel  of  the  United  States,  and  the  legal 
bushel  of  each  state  having  no  special  or  statute  bushel  of  its  own. 
The  standard  Winchester  bushel  measure  is  a  cylindrical  vessel  hav- 
ing an  outside  diameter  of  19^  inches,  an  inside  diameter  of  18£ 
inches,  and  an  inside  depth  of  8  inches.  The  standard  "  heaped  "  or 
"  coal "  bushel  of  England  was  this  measure  heaped  to  a  true  cone  6 
inches  high,  the  base  being  19£  inches,  or  equal  to  the  outside  diam- 
eter of  the  measure.  Its  ratio  to  the  even  bushel  was,  therefore,  as 
1.28,  nearly,  to  1.  The  present  "  Imperial  "  measure  of  England  has 
the  same  outside  diameter  and  the  same  depth  as  the  Winchester,  and 
an  internal  diameter  of  18.8  inches,  and  the  same  height  of  cone  is 
retained  for  forming  the  heaped  bushel.  Its  ratio,  therefore,  to  the 
even  bushel  is  a  trifle  less  than  was  that  of  the  Winchester.  In  the 
United  States  the  "  heaped  bushel  "  is  usually  estimated  at  5  even, 
pecks,  or  as  1.25  to  1  of  the  standard  even  bushel,  which,  if  taken  as 

*  By  enactment  of  the  Legislature  of  the  State  of  New  York,  this  gallon  ceased  to  be  the 
legal  gallon  of  that  State,  April  11, 1852 ;  and  the  United  States  Government  gallon,  of  231 
cubic  inches  capacity,  was  adopted  in  its  stead. 

3* 


30  DRY   MEASURE. 

the  rule,  requires  a  cone  on  the  Winchester  measure  of  5.4  inches  to 
equal  the  heaped  Winchester  bushel. 

4  gills  sm        1  pint. 

2  pints  =»         1  quart. 


4  quarts  .         .         -     ^a 


or  half  peck. 
8  quarts  .         .        =      "  1  peck. 

4  pecks  "]  fl  bushel. 

2150.42  cubic  in.       (  J  2738  cyl.  in. 

1.244456  "       ft.      r     —     S  77.7785  av'd  lbs. 
1.5844     cyl.     "      J  (.pure  water. 

Bush»l  of  the  )  (1.28  cubic  feet. 

State  of  New  York*}     =     I  2211.84  "  in. 
2816.1955  cyl.  in.     )  (  80  av'd  lbs.  pure  water. 

!  1.272  cubic  feet. 
2198      "       in. 
79.50  av'd  lbs.  pure  water. 
Heaped  Win.  bushel  >  {  2747.7  cubic  in. 

1.28— even"         "       J**      \l. 59  cubic   ft. 
Imperial  bushel  =        2218.192  "  in. 

Chaldron  =         36  Winch,  heaped  bushels. 

1  Winchester  bushel  -   i  g'^  ^Perial„bu8hel- 

}  9.3092  Wine  gallons. 

1  Imperial  bushel  =       1.0315  Winchester  bushels. 

Note.  —  The  Imperial  bushel,  mentioned  above,  is  the  present  legal  bushel  of  Great 
Britain  ;  and  the  Imperial  gallon,  mentioned  on  the  preceding  page,  is  the  present  legal 
gallon  of  Great  Britain,  for  all  liquids.  The  gallon  for  liquids  is  the  same  as  the  gallon 
for  dry  measure.  Eight  Imperial  gallons  make  one  bushel.  The  subdivisions  of  the  gal- 
lon and  the  bushel,  and  their  denominations,  are  the  same  as  in  the  Winchester  measures. 
In  Great  Britain,  in  addition,  to  the  denominations  of  dry  measure  used  in  the  United 
States,  the 


Strike, =   2  bushels. 

Coomb, =   4      " 

Quarter, =    8       " 

Wey  or  load, =  40      " 


Last, =80  bushels. 

Sack  of  corn, =   3       " 

Bole  of  corn, =    6       " 

Last  of  gunpowder,    .   .  =  42  barrels. 


*  This  bushel  ceased  to  be  the  legal  bushel  of  this  State  April  11, 1852,  and  the  United 
States  Government  bushel,  of  2160^^  cubic  inches  capacity,  was  adopted  as  the  legal 
bushel  in  its  stead. 

t  This  bushel  is  now,  January,  1852,  no  longer  the  legal  bushel  of  this  State,  and  the 
"  Winchester  bushel  is  adopted  In  its  stead. 


SECTION  II. 

MISCELLANEOUS  FACTS,    CALCULATIONS,    AND   PRACTICAL 
MATHEMATICAL   DATA. 


SPECIFIC  GRAVITIES. 
The  specific  gravity  of  a  body  is  its  weight  relative  to  the  weight 
of  an  equal  bulk  of  pure  water  at  the  maximum  density,  (39°. 83,  b. 
30  in.)  the  water  being  taken  as  1.,  a  cubic  foot  of  which  weighs 
1000  avoirdupois  ounces,  or  62£  lbs.  The  specific  gravity,  therefore, 
of  any  body  multiplied  by  1000,  or,  which  is  the  same  thing,  the  dec- 
imal being  carried  to  three  places  of  figures,  or  thousands,  as  in  the 
following  tables,  the  whole  taken  as  an  integer  equals  the  number 
of  ounces  in  a  cubic  foot  of  the  material :  multiplied  by  62.5,  or  con- 
sidered an  integer  and  divided  by  16,  it  equals  the  number  of  pounds 
in  a  cubic  foot ;  and  multiplied  by  .036169,  or  taken  as  an  integer 
and  divided  by  27648,  it  equals  the  decimal  fraction  of  a  pound  per 
cubic  inch ;  by  which,  it  is  readily  seen,  the  specific  gravity  of  a 
commodity  being  known,  its  weight  per  any  given  bulk  is  easily  and 
accurately  ascertained  ;  as,  also,  its  specific  gravity,  the  weight  and 
bulk  being  known.  The  weight  of  any  one  article  relative  to  that  of 
any  other,  is  as  its  respective  specific  gravity  to  the  specific  gravity 
of  the  other. 


METALS. 

Specific 
gravity. 

Specific 
gravity. 

Antimony,   . 

6.712 

Gold,  pure,  hammerec 

19.546 

Arsenic, 

5.810 

Iridium, 

15.363 

Bismuth, 

9.823 

Iron,  cast, 

7.209 

Bronze, 

8.700 

"    wrought,    . 

7.787 

Brass,  best, .     ■     . 

8.504 

Lead, 

11.352 

Copper,  cast, 

8.788 

Mercury,  32°,     . 

13.598 

"       wire-drawn, 

8.878 

"     '     60°,     . 

13.580 

Cadmium,    . 

8.604 

«     —39°,     . 

15.000 

Cobalt, 

7.700 

Manganese, 

8.013 

Chromium, 

5.900 

Molybdenum, 

8.611 

Glucinium, 

3.000 

Nickel, 

8.280 

Gold,  pure,  cast,  . 

19.258 

Osmium,    . 

10.000 

32 


SPECIFIC   GRAVITIES. 


Speeific 

Specific 

parity. 

gravity. 

Platinum,  cast,    . 

19.500 

Granite,  red, 

2.625 

"         hammered, 

20.337 

"         Lockport, 

2.655 

"        rolled, 

22.069 

"         Quincy, 

2.652 

Potassium,  60°,  . 

0.865 

"          Susquehanna, 

2.704 

Palladium, 

11.870 

Grindstone, . 

2.143 

Rhodium, 

11.000 

Gypsum,  opaque, 
Hone,  white, 

2.168 

Silver,  pure,  cast, 

10.474 

2.876 

"       hammered, 

10.511 

Hornblende, 

3.600 

Sodium, 

0.970 

Ivory, 

1.822 

Steel,  soft, 

7.836 

Jasper, 

2.690 

"      tempered, 

7.818 

Limestone,  green, 

3.180 

Tin,  cast, 

7.291 

"          white, 

3.156 

Tellurium, 

6.115 

Lime,  compact,    . 

2.720 

Tungsten, 

17.600 

"     foliated,      . 

2.837 

Titanium, 

4.200 

11      quick, 

0.804 

Uranium, 

9.000 

Loadstone,  . 

4.930 

Zinc,  cast, 

6.861 

Magnesia,  hyd.,    . 

2.333 

Marble,  common, 

2.686 

STONES    AND   EA 

UTIIS. 

"         white  Ital. 

2.708 

Alabaster,  white, 

2.730 

"         Rutland,  Vt., 

2.708 

"         yellow, 

2.699 

"         Parian,    . 

2.838 

Amber, 

1.078 

Nitre,  crude, 

1.900 

Asbestos,  starry, 

3.073 

Pearl,  oriental,     . 

2.650 

Borax, 

1.714 

Peat,  hard, 

1.329 

Bone,  ox,  . 

1.656 

Porcelain,  China, 

2.385 

Brick, 

1.900 

Porphyra,  red, 

2.766 

Chalk,  white,     . 

2.782 

"         green, 

2.675 

Charcoal,   . 

.441 

Quartz, 

2.647 

"         triturated, 

1.380 

Rock  Crystal, 

2.654 

Cinnabar,  . 

7.786 

Ruby, 

4.283 

Clay,          .         . 

1.934 

Stone,  common,    . 

2.520 

Coal,  bitum.  avg., 

1.270 

"      paving, 

2.416 

"     anth.      " 

1.520 

"      pumice, 

0.915 

Coral,  red, 

2.700 

"       rotten, 

1.981 

Earth,  loose, 

1.500 

Salt,  common,  solid, 

2.130 

Emery, 

4.000 

Saltpetre,  refined, 

2.090 

Feldspar, 

2.500 

Sand,  dry,  . 

1.800 

Flint,  white, 

2.594 

Serpentine, 

2.430 

"      black, 

2.582 

Shale, 

2.600 

Garnet, 

4.085 

Slate, 

2.672 

Glass,  flint, 

2.933 

Spar,  fluor, 

3.156 

"      white, 

2.892 

Stalactite,    . 

2.321 

"      plate, 

2.710 

Tale,  black, 

2.900 

"      green, 

2.642 

Topaz, 

4.011 

SPECIFIC    GRAVITIES. 


33 


Specific 

Specifio 

trr*Tity. 

gnmiy. 

SIMPLE    SUBSTANCE 

Pine,  yellow, 

.568 

neither  metallic 

;  nor  gaseous. 

Poplar,  white, 

.383 

Boron, 

1.968 

Plum,    . 

.785 

Biomine, 

2.970 

Quince, 

.705 

Carbon, 

3.521 

Spruce,  white, 

.551 

Iodine, 

4.943 

Sassafras, 

.482 

Phosphorus, 

1.770 

Sycamore, 

.604 

Selenium,    . 

.      *  .         4.320 

Walnut, 

.671 

Silicon, 

1.184 

Willow, 

.585 

Sulphur, 

1.990 

Yew,  Spanish, 

.807 

"     Dutch, 

.788 

WOODS, 

(dry.) 

Apple, 

0.793 

Highly  seasoned  Am. 

Alder, 

.800 

Ash,  white,   . 

.722 

Ash,    . 

.760 

Beech, 

.624 

Beech, 

.696 

Birch, 

.526 

Birch, 

.720 

Cedar,  . 

.452 

Box,  French, 

1.328 

Cherry, 

.606 

"     Dutch, 

.912 

Cypress, 
Elm,     . 

.441 

Cedar, 

.561 

.600 

Cherry, 

.715 

Fir,       .         . 

.491 

Chestnut,     . 

.610 

Hickory,  red, 

.838 

Cocoa, 

1.040 

Maple,  hard, 

.560 

Cork, 

.240 

Oak,  white,  uplanc 

1,       .         .687 

Cypress, 

.644 

"     James  River, 

.759 

Ebony,  American 

1.331 

Pine,  yellow, 

.541 

"        foreign, 

1.290 

"      pitch,  . 

.536 

Elm,    . 

.671 

"      white, 

.473 

Fir,  yellow, 

.657 

Poplar,  (tulip,) 

.587 

"    white, 

.569 

Spruce,  white, 

.465 

Hacmetac,   . 

.592 

Hickory,  red, 

.900 

GUMS,    FAT 

3,    &C. 

Lignum  vitse, 
Larch, 

1.333 
.544 

Asphaltum,    . 

5  .905 
'   \  1.650 

Logwood,     . 

.913 

Beeswax, 

.965 

Mahogany,  Spanis 

h,  best,      1.065 

Butter, 

.942 

<(                 tt 

com.,      .800 

Camphor, 

.988 

"         St.  Dc 

mingo,        .720 

Gamboge, 

.       1.222 

Maple,  red, 

.750 

Gunpowder,  . 

.900 

Mulberry,    . 

.897 

"            shakei 

l,       .       1.000 

Oak,  live,     . 

1.120 

solid, 

5  1.550 
*    \  1.800 

"     white, 

.785 

Orange, 

.705 

Gum,  Arabic, 

.       1.454 

Pear, 

.661 

"     Caoutchouc, . 

.933 

Pine,  white, 

.554 

"     Mastic, 

.       1.074 

34 


SPECIFIC    OrHAVITlES. 


Honey, 

Ice,     . 

Indigo, 

Lard, 

Pitch, 

Rosin, 

Spermaceti, 

Starch, 

Sugar,  dry, 

Tallow, 

Tar,    . 


LIQUIDS. 

Acid,  acetic, 

"     citric, 

"     fluoric, 

n     nitric, 

"     nitrous, 

M     sulphuric, 

' '     muriatic, 

u     silicic, 
Alcohol,  anhyd. 
"        90  °/0 
Beer, 

Blood,  human, 
Camphene,  pure, 
Cider,  whole, 
Ether,  sulph., 

"      nitric, 
Milk,  cow's, 
Molasses,  75  % 
Oils,  linseed, 

"     olive, 


"     sassafras, 
"     turpentine,  com. 
"     sperm,  pure, 
"     whale,  pTd, 
Proof  spirits, 
Vinegar, 
Water,  pure, 
"        sea, 
**       Dead  sea, 


Specific 
gravity. 

1.450 

.930 
1.009 

.941 
1.150 
1.100 

.943 
1.530 
1.606 

.938 
1.015 


1.062 

1.034 

1.060 

1.485 

1.420 

1.846 

1.200 

2.660 

.794 

.834 

1.034 

1.054 

.863 

1.018 

.715 

.908 

1.032 

1.400 

.934 

.917 

.927 

1.090 

.875 

.874 

.923 

.925 

1.025 

1.000 

1.026 

1.240 


Win 
it 

3,  champagne, 
claret, 

Specific 
gravity. 

.997 
.994 

<< 

port, 
sherry, 

.997 
.992 

ELASTIC    FLUIDS! 

The  measure  of  which  is  atmospheric  air, 
at  60°,  b.  30  in.,  its  assumed  gravity  1 ;  one 
cubic  foot  of  which  weighs  527.04  grains,  = 
.305  of  a  grain  per  cubic  inch.  It  is,  at 
this  temperature  and  density,  to  pure  water 
at  the  maximum  density,  as  .0012046  to  1, 
or  as  1  to  830.1. 


SIMPLE   OR    ELEMENTARY    GASES. 


Hydrogen, 

.0689 

Oxygen,     . 

1.1025 

Nitrogen,  . 

.9760 

Fluorine,   . 

Chlorine, 

2.470 

Carbon,  vapor  of, 

I  .422 

(tJteoretically,) 

COMPOUND   GASES. 

Ammoniacal,       .         .  .591 

Carbonic  acid,     .         .  1.525 

"        oxide,  .         .  .763 

Carbureted  hydrogen, .  .559 

Chloro-carbonic,           .  3.389 

Cyanogen,  .         .         .  1.818 

I  Muriatic  acid  gas,        .  1.247 

|  Nitrous  acid  gas,          .  3.176 

Nitrous  oxide  gas,       .  1.040 

Olefiant,      .         .         .  .982 

Phosphureted  hydrogen,  1.185 

Sulphureted          "      .  1.177 

Sfam,  212°         .         .  .484 

Smoke,  of  wood,          .  .900 

"       of  coal,             .  .102 

Vapor,  of  water,           .  .623 

"       of  alcohol,         .  1.613 

"       of  spirits  turpentine,  5.013 


WEIGHT  PER  BUSHEL  — 

BARREL GALLON,  < 

fee. 

35 

Weight  per  Bushel  (even 

Winchester)  of  different  Grains,  Seeds,  $c. 

Articles. 

it*. 

Articles. 

It.*. 

Barley,  (N.  E.  47  lbs.) 

48 

Hemp  seed, 

40 

Beans, 

64 

Oats, 

32 

Buckwheat, 

46 

Peas, 

64 

Blue-grass  seed 

14 

Rye,        .        . 

56 

Corn, 

56 

Salt,  T.  I., 

80 

Cranberries,      . 

"     boiled,      . 

56 

Clover  seed, 

60 

Timothy  seed, 

46 

Dried  Apples, 

22 

Wheat,    . 

60 

"     Peaches, 

33 

Potatoes,  h'p'd, 

60 

Flaxseed,  (N.  E.  52  lbs.) 

56 

Weight  per  Barrel  (L 

ega 

I  or  by  Usage)  of  different 

Articles. 

Flour,      .  %      . 

196  lbs. 

Cider,  in  Mass., 

32  gals. 

Boiled  Salt,      .         .         5 

180 

<« 

Soap, 

256  lbs. 

Beef,                                  I 

SOO 

it 

Raisins,   . 

112   " 

Pork,                                   5 

!00 

H 

Anchovies, 

30   " 

Pickled  Fish,   .         .        fi 

!00 

(( 

Lime, 

"         "       in  > 

Massachusetts,  } 

30 

gls. 

Ground  Plaster, 

Hydraulic  Cement,   . 

300   " 

A  Gallon  of  Train  Oil  weighs       .         .         .         7f    lbs. 
A      "       "  Molasses,  standard,  (75  per  cent.,)    11|    " 


A  Puncheon  of  Prunes, 

A  Firkin  of  Butter,  (legal,) 

A  Keg  of  powder,         .... 

A  Hogshead  of  Salt  is 

A  Perch  of  Stone  =  24i|  cubic  feet. 

A  Gallon  of  Alcohol,  90  per  cent.,  weighs 


8    bush. 

6.965  lbs. 

7.732 

8.3 

7.33 

7.71 

7.66 

7.31 

7.21 


Weight  of  Coals,  djrc,  broken  to  the  medium  size,  per  Measure  of 
Capacity. 

The  average  weight  of  Bituminous  Coals,  broken  as  above,  is  about 
62  per  cent,  that  of  a  bulk  of  equal  dimensions  in  the  solid  mass,  or 


A 
A 
A 
A 

"  Proof  Spirits, 
"  Wine,  (average,) 
"  Sperm  Oil, 
"  Whale  "    p'f 'd, 

A 

<( 

"  Olive     " 

A 
A 

11  Spirits  Turpentine, 
"  Camphene,  pure, 

1120 
56 
25 


ROPES  AND  CABLES. 


of  the  specific  gravity  of  the  article ;  that  of  Anthracite  is  about 
5  7  per  cent 


Arerage  weight  t 
peroubio  foot. )                                                               lbf. 

Average  weight  per  J 
W.  Coal  buabel.    J 

lb*. 

Anthracite,     ...          54 

Anthracite, 

86 

Bituminous,    ...          50 

Bituminous, 

80 

Charcoal,  of  pine, .        .       18.6 

Charcoal,  hard  wood, 

"        of  hardwood,.     19.02 

Coke,  best, 

.         .         .         32 

Practical  Approximate  Weight  in  Pounds  of  Various  Articles. 

Sand,  dry,  per  cubic  foot, 

95 

Clay,  compact,  per  cubic  foot, 

135 

Granite,                "      "        "           . 

165 

Lime,  quick,         "      "        "           . 

50 

Marble,                 "      "        " 

.       -    169 

Slate,                    "      "        " 

167 

Peat,  hard,           "      «        "           . 

83 

Seasoned  Beech  Wood,  per  cord,     . 
"       Yellow  Birch  Wood,  per  cord, 

5616 

4736 

"       Red  Maple  Wood,        "      "    . 

5040 

«         "    Oak  Wood,           "      " 

6200 

"       White  Pine  Wood,       "      " 

4264 

"       Hickory  Wood,             "       " 

6960 

"        Chestnut  Wood, 

U             ii 

4880 

'? 


Meadow  Hay,  well  settled,  per  cubic  foot,  8J  lbs.,  or 
240  cubic  feet  =  2000  lbs.,  or  268T^  cubic  feet 
=.  1  long  ton 

Meadow  Hay,  in  large  old  stacks,  per  cubic  foot, 

Clover  Hay,  in  settled  bulk,  "        "       " 

Corn  on  Cob,  in  crib, 
"     shelled,  in  bin, 

Wheat,  in  bin, 

Oats,  in  bin, 

Potatoes,  in  bin, 

Common  Brick,  7|  X  3f  X  2£  in. 

Front  "     8XHX2|  in. 

ROPES   AND   CABLES. 

The  strength  of  cords  depends  somewhat  upon  the  fineness  of 
the  strands ;  —  damp  cordage  is  stronger  than  dry,  and  untarred 
stonier  than  tarred ;  but  the  latter  is  impervious  to  water  and  less 
elastic. 

Silk  cords  have  three  times  the  strength  of  those  of  flax  of 
equal  circumference,  and  Manilla  has  about  half  that  of  hemp. 


it           (( 

U 

22 

u          u 

M 

45 

(«             u 

tt 

48 

({          it 

u 

25 

((          (t 

u 

38, 

»  M,  . 

. 

4500 

U    If 

. 

6185 

WEIGHT   AND   STRENGTH   OF   IKON    CHAINS. 


87 


Ropes  made  of  iron  wire  are  full  three  times  stronger  than  those 
of  hemp  of  equal  circumference. 

White  ropes  are  found  to  be  most  durable.  The  best  qualities  of 
hemp  are — 1.  pearl  gray;  2.  greenish;  3.  yellow.  A  brown  color 
has  less  strength. 

The  breaking  weight  of  a  good  hemp  rope  is  6400  lbs.  per  square 
inch,  but  no  cordage  may  be  counted  on  with  safety  as  capable  of  sus- 
taining a  weight  or  strain  above  half  that  required  to  break  it,  and 
the  weight  of  the  rope  itself  should  be  included  in  the  estimate. 

The  reliable  strength  of  a  good  hemp  cable,  in  pounds,  is  usually 
estimated  as  equal  to  the  square  of  its  circumference  in  inches  X  by 
120.  That  of  rope  X  200.  Thus,  a  cable  of  9  inches  in  circumfer- 
ence may  be  relied  on  as  having  a  sustaining  power  =  9  X  9  X  l20 
=  9720  lbs. 

The  weight,  in  pounds,  of  a  cable  laid  rope,  per  linear  foot  =  the 
square  of  its  circumference  in  inehes  X  -036,  very  nearly. 

The  weight,  in  pounds,  of  a  linear  foot  of  manilht,  rope—  the 
square  of  its  circumference  in  inches  X  -03,  very  nearly.  Thus,  a 
man  ilia  rope  of  three  inches  circumference  weighs  per  linear  foot 
3  X  3  X    03  =  -fJv  lbs.,  =  3^  feet  per  lb. 

A  good  hemp  rope  stretches  about  £,  and  its  diameter  is  diminished 
about  £  before  breaking. 

WEIGHT  AND  STRENGTH  OF  IRON  CHAINS. 


Diameter  of 

Wire 
in  Inches. 

Weight  of 

lFoot 

of  Chain. 

Breaking 
Weight 
of  Chain. 

Diameter  of 

Wire 
in  Inehes. 

Weight  of 
lFoot 
of  Chain. 

Breaking 
Weight 
of  Chain. 

lbs. 

lbs. 

lbs. 

lbs. 

A 

0.325 

2240 

1 

4.217 

26880 

i 

0.65 

4256 

« 

4.833 

32704 

A 

0.967 

6720 

f 

5.75 

38752 

1 

1.383 

9634 

if 

6.667 

45696 

7 
TS 

1.767 

13216 

i 

7.5 

51744 

i 

2.633 

17248 

« 

9.333 

58464 

* 

3.333 

21728 

1 

10.817 

65632 

38 


COMPARATIVE    WEIGHT    OF    METALS. 


Comparative  Weight  of  Metals,  Weight  per  Measure  of  Solidity,  $c 


Specific 

Ratio  of 

Pounris 

n  a  Cubic 

Iron,  wrought  or  rolled, 

Gravity. 

Comparison 

Foot. 

Inch. 

7.787 

1. 

486.65 

.28163 

Cast  Iron, 

7.209 

.9258 

450.55 

.26073 

Steel,  soft,  rolled,     . 

7.836 

1.0064 

489.75 

.28342 

Copper,  pure,    " 

8.878 

1.1401 

554.83 

.32110 

Brass,  best,        " 

8.604 

1.1050 

537.75 

.3112 

Bronze,  gun  metal,    . 

8.700 

1.1173 

543.75 

.31464 

Lead,        .... 

11.352 

1.4579 

709.50 

.4106 

TABLE, 

Exhibiting  the  Weight  in  pounds  of  One  Foot  in  Length  of  Wrought 
or  Rolled  Iron  of  any  size,  {cross  section,)  from  ft  inch  to  12  inches. 


SQUARE    BAR. 


Size 

Weight 

Size 

Weight 

Size 

Weight 

Size 

Weight 

Inches. 
1 

in 

Pounds. 

Inches. 

2& 

in 
Pound*. 

in 

Inches. 

in 
Pounds. 

Inch**. 

Pounds. 

.053 

19.066; 

41 

72.305 

71 

203.024 

1 

.211 

24 

21.120 

4| 

76.264 

8 

216.336 

1 

.475 

81 

23.292 

4ft 

80.333 

H 

230.068 

1 

.8-15 

2| 

25.560; 

5 

84.480 

84 

214.220 

1 

1.320 

2* 

27.939, 

5ft 

88.784 

81 

258.800 

1 

1.901 

3 

30.416, 

H 

93.168 

9 

273.792 

I 

2.588 

3ft 

33.010. 

58 

97.657 

9* 

289.220 

3.380 

H 

35.7041 

'  54 

102.240 

94 

305.056 

11 

4.278 

3S 

38.503, 

5& 

106.953 

0| 

321.332 

n 

5.280 

34 

41.408 

5| 

111.756 

10 

337.920 

it 

6.390 

3& 

44.418 

5* 

116.671 

10.1 

355.136 

u 

7.604 

31 

47.534 

6 

121.664 

104 

373.679 

it 

8.926 

3* 

50.756 

6i 

132.040 

10| 

390.628 

ii 

10.352 

4 

54.084 

64 

142.816 

n 

408.960 

ii 

11.883 

4ft 

57.517 

63 

151.012 

Hi 

427.812 

2 

13.520 

H 

61.055 

7 

165.632 

ii4 

117.024 

24 

15.263 

4& 

64.700 

7| 

177.672 

ill 

166. 084 

n 

17.112 

44 

68.448 

U 

190.136 

12 

486.656 

COMPARATIVE    WEIGHT   OF   METALS.  39 

To  determine  the  weight,  in  pounds,  of  one  foot  in  length,  or  of  any 
length,  of  a  bar  of  any  of  the  following  metals  of  form  prescribed,  of 
any  size,  multiply  the  weight  in  pounds,  of  an  equal  length  of  square 
rolled  iron  of  the  same  size,  (see  table  of  square  rolled  iron,)  if  the 
weight  be  sought  of 

Iron,         Round  rolled,  by 7854 

Steel,       Square      "  " 1.0064 

Round       "  " 7904 

Cast  Iron,  Square  bar,  " 9258 

"      "     Round    "  " 7271 

Copper,    Square  rolled, " 1.1401 

"  Round      "  " 8954 

Brass,       Square     "  " 1.105 

"  Round      "  " 8679 

Bronze,    Square  bar,  " 1.1173 

"  Round     "  " 8775 

Lead,       Square    "  " 1.4579 

Round     "  " 1.145 

The  weight  of  a  bar  of  any  metal,  or  other  substance,  of  any  given 
length,  of  a  flat  form,  (and  any  other  form  maybe  included  in  the 
rule,)  is  readily  obtained  by  multiplying  its  cubic  contents  (feet  or 
inches)  by  the  weight  (pounds,  ounces,  or  grains)  of  a  cubic  foot  or 
inch  of  the  article  sought  to  be  weighed ;  that  is  — 

Length  X  breadth  X  thickness  X  weight  per  unit  of  measure. 

For  the  weight  in  pounds  of  a  cubic  foot  or  inch  of  different  metals, 
see  "  Table  of  weights  of  metals  per  measure  of  solidity,  &c„" 

•  OR,  FOR  FLAT  OR  SQUARE  BARS, 

Multiply  the  sectional  area  in  inches  by  the  length  in  feet,  and  that 
product,  if  the  metal  be 

Wrought  Iron,  by 3.3795 

Cast  "      " 3.1287 

Steel,  "     .    ' 3.4 

Example.  — Required  the  weight  of  a  bar  of  #teel,  whose  length  is 
7  feet,  breadth  2£  inches,  and  thickness  £  of  an  inch. 

2.5  X  .75  X  7  X  3.4  =  44.625  lbs.     Ans. 

Example.  —  Required  the  weight  of  a  cast  iron  beam,  whose  length 
is  14  leet,  breadth  9  inches,  and  thickness  14  inch. 

14  X  9  X  1-5  X  3.1287  =  591.32  lbs.     Ans. 


40 


WEIGHT  OF   ROUND   ROLLED   IRON, 


TABLE, 

Exhibiting  the  weight  in  pounds  of  One  Foot  in  Length  of  Round  Rolled 
Iron  of  any  diameter ,  from  J  inch  to  12  inches. 


Diameter 

Weight 

Diana,  in 

Weight 

Diam.   in 

Weight 

Diam.  in 

Weight 

in  inches. 

in  lbs. 

inches. 

in  lbs. 

inches. 

in  lbs. 

inches. 

i    in  lbs. 

1 

^041 

2| 

14.975 

4| 

56,788 

71 

159.456 

1 

,165 

24 

16.688 

41 

59.900 

8 

169.S56 

1 

,373 

2| 

18.293 

±1 

63.094 

H 

180.696 

4 

.663 

23 

20.076 

5 

66.752 

4 

191.808 

I 

1.043 

2& 

21.944 

51 

69,731 

H 

208.260 

i 

1.493 

3 

23.888 

H 

73,172 

9 

215.040 

I 

2.032 

H 

25,926 

51 

76.700 

n 

227.152 

i     •! 

2,654 

n 

28.040 

54 

80.304 

% 

239.600 

I| 

3,360 

H 

30.240 

H 

84.001 

91 

252.376 

11 

4.172 

4 

32.512 

51 

87.776 

10 

266.288 

If 

5.019 

3| 

34.886 

&l 

91.634 

101 

278.924 

4 

5.972 

3| 

37.332 

6 

95,552 

m 

292.688 

it 

7.010 

*1 

39.864 

H 

103,704 

101 

306.800 

ii 

8.128 

4 

42.464 

64 

112.160 

11 

321.216 

U 

9.333 

4| 

45.174 

61 

120.960 

III 

336.004 

2 

10.616 

U 

47.952 

7 

130.048 

Hi 

351.104 

2* 

11.988 

M 

50.815 

7i 

139.544 

ill 

366.536 

1       2| 

13.440 

*h 

53.760 

74 

149.328 

12 

382.208 

To  find  the  weight  of  an  equilateral  three-sided  cast  iron  prism. 
width  of  side  m  inches   X  !•  354  X  length  in  feet  =  weight  in  lbs. 

Example.  —  A  three-&ided  cast  iron  prism  is  14  feet  m  length,  and 
the  width  of  each  Bide  is  6  inches ;  required  the  weight  of  the  prism. 

62  X  1.354  X  14  =  682.4  lbs.     Ans. 
To  find  the  weight  of  an  equilateral  rectangular  cast  iron  prism. 

width  of  side  in  inches"  X  3.128  X  length  in  feet  =  weight  in  lbs. 
To  find  the  weight  of  an  equilateral  five-sided  cast  iron  prism. 

width  of  side  in  inches2  X  5.381  X  length  in  feet  =  weight  in  lbs. 
To  find  the  weight  of  an  equilateral  six-sided  cast  iron  prism. 

width  of  side  in  inches2  X  8.128  X  length  in  feet  =  weight  in  lbs. 
To  find  the  weight  of  an  equilateral  eight-sided  cast  iron  prism. 

width  of  side  in  inches2  X  15-1  X  length  in  feet  =  weight  in  lbs. 

To  find  the  weight  of  a  cast  iron  cylinder. 

diameter  in  inches2  X  2.457  X  length  in  feet  =  weight  in  lbs. 

In  a  quantity  of  cast  iron  weighing  125  lbs.,  how  many  cubic 
inches  ? 

By  tabular  weight  per  cubic  inch  — 

125  -j-  .26073  =*  479.4  cubic  inches.    An*. 


RELATING   TO   CAST   IRON.  41 

Or,  by  tabular  weight  per  cubic  foot  — 

450.55  :  1728  :  :  125  :  479.4  cubic  inches.    Ans. 
How  many  cubic  inches  of  copper  will  weigh  as  much  as  479.4 
cubic  inches  of  cast  iron  ? 

By  tabular  weight  per  cubic  inch  — 

.3211  :  .2G073  :  :  479.4  :  389.27  cubic  inches.     Ans. 
Or,  by  specific  gravities  — 

8.878  :  7.209  :  :  479.4  :  389.27  cubic  inches.    Ans. 
Or,  by  tabular  ratio  of  weight  — 
.9258 
479.4  x  1J401  -  389.28. 

A  cast  ircn  rectangular  weight  is  to  be  constructed  having  a 
breadth  of  4  inches  and  a  thickness  of  2  inches,  and  its  weight  is 
to  be  18  lbs. ;  what  must  be  its  length  ? 
18 
4X2X.20073=as8-63inches-     Ans' 
A  cast  iron  cylinder  is  to  be  2  inches  in  diameter,  and  is  to  weigh 

6  lbs. ;  what  must  be  its  length  ? 

.26073  X  .7854  =.2047  lb.  —  weight  of  1  cyl.  inch,  then 
/» 

2^X.2047=7-327iDCheS-     AnS- 
A  cast  iron  cylinder  is  to  weigh  6  lbs.,  and  its  length  is  to  bo 
7.327  inches  ;  what  must  be  its  diameter? 

**(  7.327  X  .2047  )  =  2  inches.     Ans.    ' 

A  cast  iron  weight,  in  the  form  of  a  prismoid,  or  the  frustrum  of 
a  pyramid,  or  the  frustrum  of  a  cone,  is  to  be  constructed  that  will 
weigh  14  lbs.,  and  the  area  of  one  of  the  bases  is  to  be  16  inches, 
and  that  of  the  other  4  inches ;  what  must  be  the  length  of  tho 
weight  ? 

14 

^/IG  X4  =  8and8-fl6-f-4-h3  =  9.33,  and  9  33  x  26073 
■be  5.75  inches.     Ans.  ' 

Note.  —  For  Rules  in  detail  pertaining  to  the  foregoing,  see  Geometrt,  Mensuration 
of  superficies  —  of  solids. 

A  model  for  a  piece  of  casting,  made  of  dry  white  pine,  weighs 

7  lbs. ;  what  will  the  casting  weigh,  if  made  of  common  brass  ? 
By  specific  gravities  — 

.554  :  8.604  :  :  7  :  108.71  lbs.    Ans. 

Note.  —  As  the  specific  gravity  of  the  substance  of  which  the  model  is  composed  must 
generally  remain  to  some  extent  uncertain,  calculations  of  this  kind  can  only  be  relied  on 
as  approximate. 

4* 


TABLE 


Exhibiting  the  Weight  of  One  Foot  in  Length  of  Flat,  Rolled  Iron; 
Breadth  and  Thickness  in  Inches,  Weight  in  Pounds. 


Br.  and  Th. 

Wei't. 

Br.  and  Th. 

Wei't. 

Br.  and  Th. 

Wei't. 

Br.  and  Th. 

WeFt. 

inch. 

lbs. 

inch. 

lbs. 

inch. 

lbs. 

inch. 

lbs. 

h  by 

1 

.211 

H  by     | 

3.696 

11  by     h 

2.957 

2J  by    & 

3.591 

| 

.422 

1 

4.224 

1 

3.696 

| 

4.488 

i 

.634 

n 

4.752 

i 

4.435 

% 

5.386 

t    by 

\ 

.264 

n  by     | 

.581 

1 

6.175 

i 

6.284 

I 

.528 

i 

1.161 

1 

5.914 

1 

7.181 

I 

.792 

§ 

1.742 

n 

6.653 

1| 

8.079 

I 

1.056 

I 

2.323 

n 

7.393 

14 

8.977 

1    by 

1 

.316 

1 

2.904 

n 

8.132 

l| 

9.874 

i 

.633 

1 

3.485 

n 

8.871 

1| 

10.772 

1 

.950 

I 

4.066 

n 

9.610 

H  by     I 

.960 

I 

1.267 

1 

4.647 

n  by    4 

.792 

i 

1.901 

| 

1.584 

n 

5.228 

1 

1.584 

1 

2.851 

i    by 

i 

.369 

n 

5.808 

1 

2.376 

| 

3.802 

i 

.739 

U  by     J 

.634 

i 

3.168 

4.752 

i 

1.108 

i 

1.267 

§ 

3.960 

| 

5.703 

i 

1.478 

i 

1.901 

\ 

4.752 

| 

6.653 

i 

1.848 

1 

2.534 

i 

5.544 

1 

7.604 

i 

2.218 

1 

3.168 

i 

6.336 

n 

8.554 

1    by 

i 

.422 

1 

3.802 

n 

7-129 

H 

9.505 

i 

.845 

I 

4.435 

n 

7.921 

n 

10.455 

l 

1.267 

1 

5.069 

n 

8.713 

n 

11.406 

i 

1.690 

11 

5.703 

H 

9.505 

n 

12.356 

i 

2.112 

H 

6.337 

If 

10.297 

n 

13.307 

1 

2.534 

11 

6.970 

11 

11.089 

21  by     J 

1.003 

i 

2.957 

H  by    i 

.686 

2     by     * 

.845 

i 

2.006 

H    by 

i 

.475 

i 

1.373 

l 

1.690 

3.010 

I 

.950 

i 

2.069 

1 

2.534 

i 

4.013 

I 

1.425 

I 

2.746 

i 

3.379 

1 

6.016 

i 

1.901 

i 

3.432 

4.224 

i 

6.019 

i 

2.376 

i 

4.119 

I 

5.069 

i 

7.023 

:i 

2.851 

1 

4.805 

i 

5.914 

i 

8.026 

I 

3.326 

l 

5.492 

1 

6.759 

n 

9.029 

i 

8.802 

U 

6.178 

IS 

7.604 

n 

10.032 

U    by 

j 

.528 

n 

6.864 

H 

8.449 

ii 

11.036 

i 

1.056 

IS 

7.551 

l| 

9.294 

n 

12.089 

.2 

1.684 

n 

8.237 

U 

10.138 

n 

13.042 

1 

2.112 

H  by     J 

.739 

2»   by     J 

.898 

n 

14.046 

1 

2.640 

i 

1.478 

\ 

1.796 

2 

16.052 

1 

8.168 

i 

2.218 

I 

2.698 

2*  by     i 

1.056 

WEIGHT   OF   FLAT,    ROLLED   IRON. 

TABLE.  —  Continued. 


43 


Br.  andTh 

Weight. 

Br.  and  Th 

Weight. 

Br.  and  Th. 

Weight. 

Br.  and  Th. 

Weight. 

inch. 

lbs. 

inch. 

I*. 

inch. 

lbs. 

inch. 

lbs. 

24  by  i 

2.112 

21  by  1| 

16.264 

34  by  | 

6.865 

8|  by  1| 

20.694 

I 

3.168 

n 

17.426 

1 

8.238 

U 

22.178 

h 

4.224 

2 

18.587 

I 

9.610 

n 

23.762 

1 

5.280 

at 

19.749 

1 

10.983 

2 

25.347 

1 

6.336 

24 

20.911 

« 

12.356 

24 

28.515 

i 

7.393 

2}   by  i 

1.214 

li 

13.729 

2i 

81.683 

1 

8.449 

i 

2.429 

18 

15.102 

21 

34.851 

n 

9.505 

s 

3.644 

li 

16.475 

4  by  J 

1.690 

14 

10.561 

i 

4.858 

U 

17.848 

4 

3.379 

18 

11.617 

§ 

6.073 

11 

19.221 

i 

6.759 

li 

12.673 

1 

7.287 

11 

20.594 

1 

10.139 

1| 

13.729 

I 

8.502 

2 

21.967 

l 

13.518 

11 

14.785 

1 

9.716 

21 

24.713 

14 

16.898 

H 

15.841 

1* 

10.931 

2i 

27.459 

li 

20.277 

2 

16.898 

14 

12.145 

3i  by  J 

1.478 

it 

23.657 

2|  by  k 

1.109 

11 

13.360 

i 

2.957 

2 

27.036 

4 

2.218 

11 

14.574 

§ 

4.436 

24 

30.416 

1 

8.327 

if 

15.789 

i 

5.914 

2i 

33.795 

4 

4.436 

11 

17.003 

& 

7.393 

21 

37.175 

I 

5.545 

n 

18.218 

1 

8.871 

3 

40.555 

i 

6.653 

2 

19.432 

$ 

10.350 

34 

43.934 

I 

7.762 

2J 

20.647 

1 

11.828 

41  by  ft 

1.795 

1 

8.871 

2* 

21.861 

H 

13.307 

k 

8.591 

U 

9.980 

3  by  i 

1.267 

ii 

14.785 

i 

7.181 

u 

11.089 

4 

2.535 

ii 

16.264 

1 

10.772 

If 

12.198 

& 

3.802 

ii 

17.748 

1 

14.363 

1J 

13.307 

i 

5.069 

it 

19.221 

14 

17.954 

it 

14.416 

| 

6.337 

li 

20.700 

li 

21.544 

15 

15.525 

| 

7.604 

li 

22.178 

H 

25.135 

li 

16.034 

J 

8.871 

2 

23.657 

2 

28.726 

2 

17.742 

1 

10.139 

21 

26.614 

24 

32.317 

2J 

18.851 

li 

11.406 

2i 

29.571 

2i 

35.908 

2|  by  J 

1.162 

H 

12.673 

2% 

32.528 

21 

39.498 

4 

2.323 

11 

13.941 

3|  by  | 

1.584 

3 

43.089 

1 

3.485 

ii 

15.208 

1 

3.168 

34 

46.680 

4 

4.647 

it 

16.475 

i 

4.752 

3i 

50.271 

s 

5.808 

11 

17.743 

i 

6.337 

4i  by  i 

3.802 

I 

6.970 

ii 

19.010 

I 

7.921 

i 

7.604 

I 

8.132 

21 

20.277 

1 

9.505 

1 

11.406 

1 

9.294 

2 

22.812 

£ 

11.089 

1 

15.208 

11 

10.455 

24 

25.345 

1 

12.673 

14 

19.010 

li 

11.617 

31  by  a 

1.373 

li 

14.257 

li 

22.812 

11 

12.779 

4 

2.746 

U 

15.842 

11 

26.614 

li 

13.940 

1 

4.119 

If 

17.426 

2 

30.416 

It 

15.102 

i 

5.492 

li 

19.010 

24 

34.218 

44 


WEIGHT   OF   FLAT,    ROLLED   IRON. 
TABLE.  —  Continued. 


Br.  and  Th. 

Weight. 

Br.  and  Th.j  Weight. 

Br.  and  Th. 

Weight. 

Br.  and  Th. 

Weight. 

inch. 

lbs. 

inch.           lbs. 

inch. 

lbs. 

inch. 

lbs. 

44  by  24 

38.020 

4|  by  3  ,48.158 

51  by    | 

13.307 

54  by  2 

37.175 

2| 

U.822 

3i;52.172 

1 

17'.  743 

24 

46.469 

3 

46.624 

3^  56.185 

li 

22.178 

3 

55.762 

H 

49.426 

5     by    i    4.224 

u 

26.614 

51  by    i 

4.858 

H 

53.228 

4    8.449 

11 

31.049 

4 

9.716 

4|  by    i 

4.013 

112.673 

2 

35.485 

I 

14.674 

4 

8.026 

1    16.898 

•     21 

39.921 

1 

19.482 

\ 

12.040 

14  '21. 122 

2* 

44.856 

Vi 

24.290 

1 

16.053 

14  25.347 

3 

53.228 

14  29.146 

14 

20.006 

11  29.571 

54  by    1 

4.647 

11 

34.007 

1| 

24.079 

2    33.795 

4 

9.294 

2 

38.865 

1| 

28.092 

2\  38.020 
2|  42.244 

1 

13.941 

24 

43.723 

2 

32.106 

1 

18.587 

24 

48.581 

H 

36.119 

3    46.469 

li 

23.234 

3 

58.297 

24 

40.132 

64.  by    4    4.436 

Id 

27.881 

6    by    4 

5.069 

n 

44.145 

4!  8.871 

13 

32.528 

WEIGHT  OF  METALS  IN  PLATE. 
The  weight  of  a  square  foot  one  inch  thick  of 


Malleable  Iron 

=  40.554  lbs. 

Com.  plate  " 
Cast  Iron 

=  37.761    " 

=  37.546    " 

Copper,  wrought     . 

=  46.240   " 

"       com.  plate . 

=  45.312    " 

Brass,  plate,  com.   . 

=  42.812    " 

Zinc,  cast,  pure 

=  35.734    " 

"     sheet     . 

=  37.448   " 

Lead,  cast 

=  59.125    " 

And  for  any  other  thickness,  greater  or  less,  it  is  the  same  in  pro- 
portion ;  thus,  a  square  foot  of  sheet  copper  ^  of  an  inch  thick 
=  46.24-^16  =  2.89  lbs.  And  5  square  feet  at  that  thickness 
=  2.89  X  5=  14.45  lbs.,  &c.  So,  too,  5  square  feet  at  2J  inches 
thickness  =  46.24  X  2.5  X  5  =  578  lbs. 


AMERICAN    WIRE   GAUGE. 


45 


THE  AMERICAN  WIRE  GAUGE. 

The  American  Wire  Gauge  was  prepared  by  Messrs.  Brown  and 
Sharp,  manufacturers  of  machinists'  tools,  Providence,  R.  I.  It  is 
graded  upon  geometrical  principles,  is  rapidly  becoming  the  stand- 
ard gauge  with  manufacturers  of  wire  and  plate  in  the  United 
States,  and  cannot  fail  to  supersede  the  use  of  the  Birmingham 
Gauge  in  this  country. 


TABLE 


Showing  the  Linear  Measures  represented  by  Nos.  American  Wire 

Gauge  and  Birmingham  Wire  Gauge,  or  the  values  of 

the  Nos.  in  the  United-States  Standard  Inch. 


American 

Birm. 

American 

Birm. 

American 

Birm. 

American 

Birm. 

No. 

Gauge. 

Gauge. 

No. 

Gauge. 

Gauge 

No. 

Gauge. 

Gauge. 

No. 

Gauge. 

Gauge. 

'inch. 

Inch. 

Inch. 

Inch. 

Inch. 

Inch. 

Inch. 

Inch. 

0000 

.46000 

.454 

8 

.12849 

.165 

19 

.03589 

.042 

30 

.01003 

.012 

000 

.40964 

.425 

9 

.11443 

.148 

20 

.03196 

.035 

31 

.0089^3 

.010 

00 

.36480.380 

10 

.10189 

.134 

21 

.02846 

.032; 

32 

.00795 

.009 

0 

.32486.340 

11 

.09074 

.120 

22 

.02535 

.028 

33 

.00708 

.008 

1 

.28930.300 

12 

.08081 

.109 

23 

.02257 

.025 

34 

.00630 

.007 

2 

.25763.284 

13 

.07196 

.095 

24 

.02010 

.022 

35 

.00561 

.005 

3 

.22942 

.259 

14 

.06408 

.083 

25 

.01790 

.020 

36 

.00500 

.004 

4 

.20431 

.238 

15 

.05707 

.072 

26 

.01594 

.018! 

37 

.00445 

5 

.18194 

.220 

16 

.05082 

.065 

27 

.01419 

.016 

38 

.00396 

6 

.16202 

.203 

17 

.04526 

.058 

28 

.01264 

.014 

39 

.00353 

7 

.14428 

.180 

18 

.04030 

.049 

29 

.01126 

.013 

40 

.00314 

Thus  the  diameter  or  size  of  No.  4  wire,  American  gauge,  is 
0.20431  of  an  inch;  Birmingham  gauge,  0.238  of  an  inch:  so  the 
thickness  of  No.  4  plate,  American  gauge,  is  0.20431  of  an 
inch ;  Birmingham  gauge,  0.238  of  an  inch ;  and  so  for  the  other 
Nos.  on  the  gauges  respectively. 


TABLE 

Showing  the  Number  of  Linear  Feet  in  One  Pound,  Avoirdupois,  of 
Different  Kinds  of  Wire  :  Sizes  or  Diameters  corre- 
sponding to  Nos.  American  Wire-gauge. 


No. 

Iron. 

Copper. 

Brass. 

No. 
19 

Iron. 

Copper. 

Brass. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

0000 

1.7834 

1.5616 

1.6552 

293.00 

256.57 

271.94 

000 

2.2488 

1.9692 

2.0872 

20 

396.41 

347.12 

367.92 

00 

2.8356 

2.4830 

2.6318 

21 

465.83 

407.91 

432.35 

0 

3.5757 

3.1311 

3.3187 

22 

587.35 

514.32 

545.13 

1 

4.5088 

3.9482 

4.1847 

23 

.  740.74 

648.63 

687.50 

2 

5.6854 

4.9785 

5.2768 

'24 

934.03 

817.89 

866.90 

3 

7.1695 

6.2780 

6.6542 

25 

1177.7 

1031.3 

1093.0 

4 

9.0403 

7.9162 

8.3906 

26 

1485.0 

1300.4 

1378.3 

5 

11.400 

9.9825 

10.581 

27 

1872.7 

1639.8 

1738.1 

6 

14.375 

12.588 

13.342 

28 

2361.4 

2067.8 

2191.7 

7 

18.127 

15.873 

16.824 

29 

2977.9 

2607.6 

2763.8 

8 

22.857 

20.015 

21.214 

30 

3754.8 

3287.9 

3484.9 

9 

28.819 

25.235 

26.748 

31 

4734.2 

4145.5 

4394.0 

10 

36.348 

31.828 

33.735 

32 

5970.6 

5221.2 

5541.4 

11  45.829 

40.131 

42.535 

33 

7528.1 

6592.0 

6987.0 

12 

57.790 

50.604 

53.636 

34 

9495.6 

8314.9 

8813.1 

13 

72.949 

63.878 

67.706 

35 

11972 

10483 

11111 

14 

91.861 

80.439 

85.258 

36 

15094 

13217 

14009 

15 

115.86 

100.75 

107.53 

37 

19030 

16664 

17662 

it;  U6.10 

127.94 

135.60 

38 

24003 

21018 

22278 

17184.26 

168.35 

171.02 

39 

30266 

26503 

28091 

181232.34 

203.45 

215.64 

40 

38176 

33342 

35432 

Notk.  —  In  tliis  TABLE  the  iron  and  copper  employed  are  supposed  to  be 
nearly  pure.  The  ipedJta  ff&rlty  of  the  former  was  taken  at  7.774  j  that  of  the 
latter,  at  &.S78.    The  specihe  gravity  ol  the  brass  wus  taken  at  S.37G. 


WIRE    AND    WIRE    GAUGES. 


47 


To  find  the  number  of  feel  in  a  pound  of  wire  of  any  material  not 
given  in  the  table,  of  any  size,  American  gauge,  its  specific 
gravity  being  known. 

Rule.  — Multiply  the  number  of  i'eet  in  a  pound  of  iron  wire  of 
the  same  size  by  7.774,  and  divide  the  product  by  the  specific  grav- 
ity of  the  wire  whose  length  is  sought;  or  ordinarily,  for  steel  wire, 
multiply  the  number  of  feet  in  a  pound  of  iron  wire  of  the  same 
size  by  0.991. 

To  find  the  number  of  feet  in  a  pound  of  wire  of  any  given  No., 
Birmingham  gauge. 
Rule.  —  Multiply  the  number  of  feet  in  a  pound  of  the  same 
kind  of  wire,  same  No.,  American  gauge,  by  the  size,  American 
gauge,  and  divide  the  product  by  the  size,  Birmingham  gauge. 

Example.  —  In  a  pound  of  copper  wire  No.  16,  American 
gauge,  there  are  127.94  feet :  how  many  feet  are  there  of  the  same 
kind  of  wire,  same  No.,  Birmingham  gauge  ? 

(127.94  X  -05082)  -^  .065  =  100.03.     Ans. 

To  find  the  weight  of  any  given  length  of  wire  of  any  given  No.  or 
size,  American  gauge,  or  the  length  in  any  given  weight,  by  help 
of  the  foregoing  table. 
Example.  —  Required  the  weight  of  600  feet  of  No.  18  iron 

wire. 

600  -f-  232.34  =  2.5822  lbs.  =  2  lbs.  9±  oz.,  nearly.     Ans. 

Example.  — Required  the  length  in  feet  of  2^  lbs.  of  No.  31 
brass  wire. 

4394  X  2.5  =  10985.     Ans. 

Characteristics  of  Alloys  of  Copper  and  Zinc  —  Brass. 


Parts  by  Weight. 

Specific 
Gravity. 

Color. 

Denomination. 

Copper. 

Zinc. 

83 
80 
741 

491 
33 

17 

20 

25£ 
34 
501 
67 

8.415 
8.448 
8.397 
8.299 
8.230 
8.284 

Yellowish  Red. 

u              a 

Pale  yellow. 
Full 

M                (4 

Deep       " 

Bath  Metal. 
Dutch  Brass. 
Rolled  Sheet  Brass. 
English  Sheet  Brass. 
German  Sheet  Brass. 
Watchmaker's  Brass. 

Note.  — To  alloys  of  copper  and  zinc,  generally,  there  is  added  a  small 
quantity  of  lead,  which  renders  them  the  better  adapted  for  turning2  planing, 
or  riling ;  and,  for  the  same  reason,  to  alloys  of  copper  and  tin,  there  is  usually 
added  a  small  quantity  of  zinc  (see  Alloys  and  Compositions), 


TABLE 


Showing  the  Weight  of  One  Square  Foot  of  Rolled  Metals,  thickness 
corresponding  to  Nos.,  American  Wire-gauge. 


Thickness. 

Iron. 

Steel. 

Copper. 

Brass. 

Lead. 

Zinc. 

No. 

Pounds. 

Pounds. 

Pounds. 

Pounds. 

Pounds. 

Pounds. 

1 

10.849 

10.999 

13.109 

12.401 

17.102 

10.833 

2 

9.6611 

9.7953 

11.674 

11.043 

15.228 

9.6466 

3 

8.6032 

8.7227 

10.396 

9.8340 

13.562 

8.5903 

4 

7.6616 

7.7680 

9.2578 

8.7576 

12.078 

7.6501 

5 

6.8228 

6.9175 

8.2442 

7.7988 

10.755 

6.8126 

6 

6.0758 

6.1601 

7.3416 

6.9450 

9.5779 

6.0667 

7 

5.4105 

5.4856 

6.5377 

6.1845 

8.5291 

5.4024 

8 

4.8184 

4.8853 

5.8222 

5.5077 

7.5957 

4.8112 

9 

4.2911 

4.3507 

5.1851 

4.9050 

6.7645 

4.2847 

10 

3.8209 

3.8740 

4.6169 

4.3675 

6.0233 

3.8151 

11 

3.4028 

3.4501 

4.1117 

3.8896 

5.3642 

3.3977 

12 

3.0303 

3.0720 

3.6616 

3.4638 

4.7770 

3.0257 

13 

2.6985 

2.7360 

3.2607 

3.0845 

4.2539 

2.6934 

14 

2.4035 

2.4365 

2.9042 

2.7473 

3.7889 

2.3999 

15 

2.1401 

2.1698 

2.5829 

2.4463 

3.3737 

2.1369 

16 

1.9058 

1.9322 

2.3028 

2.1784 

3.0043 

1.9029 

17 

1.6971 

1.7207 

2.0506 

1.9399 

*  2.6753 

1.6945 

18 

1.5114 

1.5324 

1.8263 

1.7276 

2.3826 

1.5091 

19 

1.3459 

1.3646 

1.6263 

1.5384 

2.1217 

1.3439 

20 

1.1985 

1.2152 

1.4482 

1.3700 

1.8893 

1.1967 

21 

1.0673 

1.0821 

1.2897 

1.2300 

1.6768 

1.0657 

22 

.95051 

.96371 

1.1485 

1.0865 

1.4984 

.94908 

23 

.84641 

.85815 

1.0227 

.96749 

1.3343 

.84514 

24 

.75375 

.76422 

.91078 

.86158 

1.1882 

.75262 

25 

.67125 

.68057 

.81109 

.76728 

1.0582 

.67024 

2G 

.59775 

.60605 

.72228 

.68326 

.94229 

.59685 

27 

.53231 

.53970 

.64345 

.60846 

.83913 

.53151 

28 

.47404 

.48062 

.57280 

.54185 

.74728 

.47333 

29 

.42214 

.42800 

.51009 

.48242 

.66546 

.42151 

80 

.37594 

..'{si  16 

.4  5426 

.42972 

.59263 

.37538 

Notk.  —  In  calculating  the  foregoing  tablk,  tho  specific  gravities  were 


taken  as  follows:  viz.,  iron, 
lead,  11.350;  Zinc,  7.189. 


7.200}  steel,  7.300;  copper,  8,700  j  brass,  8.ii30j 


TIN   PLATES. 


49 


TIN  PLATES. 


Brand 

Marks. 

Size  of 

No.  of 

Sheets  in 

Sheets 

Inches. 

in  Box. 

IC 

14X14 

200 

IC 

14  X  10 

225 

HC 

14X10 

225 

HX 

14  X  10 

225 

IX 

14  X  10 

225 

IXX 

14  X  10 

225 

IXXX 

14  X  10 

225 

IXXXX 

14  X  10 

225 

IX 

14  X  14 

200 

IXX 

14X14 

200 

DC 

17X12£ 

100 

DX 

17X12* 

100 

DXX 

17X12* 

100 

DXXX 

17X12* 

100 

DXXXX 

17X12* 

100 

SDC 

15XH 

200 

SDX 

loXH 

200 

140 
112 
119 
147 
140 
161 
182 
203 
174 
200 
105 
12G 
147 
168 
189 
168 
189 


Brand  Marks. 


Size  of 
Sheets  in 
Inches. 


No.  of 
Sheets 
in  Box. 


15  X 

15  X 
15  X 
14  X 


SDXX 
SDXXX 
SDXXXX 
TT 

IC12X 
"  1X12  X 

IXXI12X 

"     IXXX!  12  X 

«IXXXXll2X 

ICJ20  X 

"  IX|20  X 

IXX  20  X 

«      IXXX  20  X 

"  IXXXX  20  x 

Ternes   IC  20  X 

"         IX;20  X 


200 
200 
200 
225 
225 
225 
225 
225 
225 
112 
112 
112 
112 
112 
112 
112 


Net 
Weight 
in  lbs. 


210 
231 
252 
112 
119 
147 
168 
189 
210 
112 
140 
161 
182 
203 
112 
140 


Note.  —  The  above  table  includes  all  the  regular  sizes  and  qualities  of 
tin  plates,  except  "  wasters."  Other  sizes,  such  as  10  X  10,  11  X  H»  13  X  13, 
&c,  of  the  different  brands,  are  often  imported  into  the  United  States  to 
order. 

Common  English  Sheet  Iron,  Nos.  10  to  28,  Birmingham  gauge, 
widths  from  24  to  36  inches. 

R.  G.  Sheet  Iron,  Nos.  10  to  30,  Birmingham  gauge,  widths  from 
24  to  36  inches. 

American  Puddled  Sheet  Iron,  Nos.  22  to  28,  Birmingham 
gauge,  widths  from  24  to  36  inches. 

Russia  Sheet  Iron,  Nos.  16  to  8  inclusive,  Russia  gauge,  sheets 
28  X  56  inches. 

Sheet  Zinc,  Nos.  16  to  8,  Liege  gauge,  widths  from  24  to  40 
inches ;  length  84  inches. 

Copper  Sheathing,  14  X  48  inches,  14  to  32  oz.  (even  numbers), 
per  square  foot. 

Yellow  Metal,  in  sheets,  48  X  14  inches,  14  to  32  oz.  (even  num- 
bers), per  square  foot. 
5 


TABLE 

Showing  the  Capacity,  in    Wine   Gallons,  of  Cylindrical  Cans,  of 
different  diameters,  at  One  Inch  depth.     Diameter  in  Inches. 


Diam'r. 

Gallons. 

Diam'r. 

Gallons. 

Diam'r. 

Gallons. 

Diam'r. 

Gallons. 

inches. 

inches. 

inches. 

inches. 

6 

.1224 

124 

.5102 

184 

1.104 

24| 

2.083 

64 

.1328 

124 

.5313 

l8i 

1.195 

2.-> 

2.125 

64 

.1437 

12| 

.5527 

i'.>- 

1.227 

254 

2.107 

61 

.1549 

13 

.5746 

104 

1.260 

254 

2.211    ! 

7 

.1666 

134 

.5969 

194 

1.293 

251 

2.254   1 

74 

.1787 

134 

.011)7 

195 

L.326 

26 

2.298 

74 

.1913 

13| 

.6428 

20 

1.360 

204 

2.343 

71 

.2042 

14 

.6604 

204 

1.394 

204 

2.388 

8 

.2176 

Hi 

.6904 

204 

1.429 

261 

2.433 

84 

.2314 

144 

.7149 

201 

1.404 

27 

2.479 

84 

.2467 

141 

.7397 

21 

1.499 

274 

2.524 

81 

.2603 

15 

.7650 

214 

1.535 

274 

2.571 

9 

.2754 

154 

.7907 

214 

1.572 

271 

2.518 

94 

.2909 

154 

.8169 

21| 

1.608 

28 

2.000 

94 

.3069 

151 

.8434 

22 

1.646 

284 

2.713 

91 

.3233 

16 

.8704 

224 

1.683 

284 

2.702 

10 

.3400 

164 

.8978 

224 

1.721 

281 

2.810 

104 

.3572 

164 

.9257 

221 

1.760 

29 

2.859 

104 

.3749 

161 

.9539 

23 

1.799 

294 

2.909 

101 

.3929 

17 

.9826 

234 

1.837 

291 

3.009 

11 

.4114 

174 

1.0120 

234 

1.877 

30 

3.060 

11*. 

.4303 

174 

1.0410 

231 

1.918 

304 

3.163 

114 

.4497 

171 

1.0710 

24 

1.958 

31 

3.264 

111 

.4694 

18 

1.1020 

244 

1.999 

314 

3.374 

12 

.4896 

184 

1.1320 

244 

2.041 

32 

3.482 

Applications  of the  foregoing  table. 

Example. — A  cylindrical  can  is  114  inches  in  diameter,  and  its 
depth  is  18f  inches  ;  required  its  capacity. 

.4303  X  18f  =  8  gallons.     Ans. 

Example.  —  The  diameter  of  a  can  containing  oil  is  264  inches,  and 
the  oil  is  144  inches  in  depth.     How  many  gallons  are  there  of  the  oil  ? 

2.388  X  144  =  34.6  gallons.     Ans. 

Example.  —  A  can  is  to  be  constructed  that  will  hold  just  36  gal- 
lons, and  its  diameter  is  to  be  18  inches ;  what  must  be  its  depth  7 

36  4-  1 .  102  =  32|  inches.     Ans. 


CAPACITY   OF   CYLINDRICAL  CANS.  51 

» 

Example.  —  A  cylindrical  can  is  to  be  constructed  that  shall  have 
a  depth  of  15  inches  and  a  capacity  of  just  5  gallons ;  what  must  be 
its  diameter? 

5  -T- 15  =  .3333  =  capacity  of  can  in  gallons  for  each  inch  of  depth ; 
and  against  .3333  gallon  in  the  table,  or  the  quantity  in  gallons 
nearest  thereto,  is  10  inches,  the  required,  or  nearest  tabular  diam- 
eter.    Ans. 

Note.  —  The  table  is  not  intended  to  meet  demands  of  the  nature  of  the  one  contained  in 
the  last  example,  with  accuracy,  unless  the  fractional  part  of  the  diameter,  if  there  be  a 
fractional  part,  is  i,  i  or  %  inch.  As,  however,  the  diameter  opposite  the  tabular  gallon 
nearest  the  one  sought,  even  at  its  greatest  possible  remove,  can  be  but  about  i  inch  from 
the  diameter  required,  we  can,  by  inspection,  determine  the  diameter  to  be  taken,  or  true 
answer  to  the  Inquiry,  sufficiently  near  for  practical  purposes,  be  the  fraction  what  it  may. 
Or,  to  throw  the  demand  into  a  mathematical  formula  :  As  the  tabular  gallon  nearest  the 
one  sought  is  to  the  diameter  opposite,  so  is  the  tabular  gallon  required  to  the  required 
diameter,  nearly.    Thus,  in  answer  to  the  last  query, 

.3400  :  10  : :  3333  :  9.8  inches,  the  required  or  true  diameter,  nearly. 

For  a  mathematical  formula  strictly  applicable  to  this  question,  see  Gauging 

Or,  for  a  formula  more  strictly  geometrical,  we  have 

.Capacity  X  231        .. 

aY  — —^ =  diameter. 

W  Depth  X  -7864 

The  true  diameter,  therefore,  for  the  supposed  can,  i3 


52  WEIGHT   OF   PIPES. 

WEIGHT  OF  PIPES. 

The  weight  of  one  foot  in  length  of  a  pipe,  of  any  diametei 
and  thickness,  may  be  ascertained  by  multiplying  the  square  of  its 
exterior  diameter,  in  inches,  by  the  weight  of  12  cylindrical  inches  of 
the  material  of  which  the  pipe  is  composed,  and  by  multiplying  the 
square  of  its  interior  diameter,  in  inches,  by  the  same  factor  and  sub- 
tracting the  product  of  the  latter  from  that  of  the  former, — the 
remainder  or  difference  will  be  the  weight.  This  is  evident  from  the 
fact  that  the  process  obtains  the  weight  of  two  solid  cylinders  of  equal 
length,  (one  foot,)  the  diameter  of  one  being  that  of  the  pipe,  and  the 
other  that  of  the  vacancy,  or  bore.  For  very  large  pipes,  the  dimen- 
sions may  be  taken  in  feet,  and  the  weight  of  a  cylindrical  foot  of  the 
material  used  as  the  factor,  or  multiplier,  if  desired. 

The  weight  of  12  cylindrical  inches  (length  1  foot,  diameter  1  inch) 
of 

Malleable  Iron  =  2.6543  lbs. 

Cast  Iron  =2.4573    " 

Copper,  wrought,  =  3.0317   " 

Lead  "  =3.8697   " 

Cast  Iron—  1  cyl.  foot—    =  353.86   " 

Therefore  —  Example.  —  Required  the  weight  of  a  copper  pipe 
whose  length  is  5  feet,  exterior  diameter  3^  inches,  and  interior 
diameter  3  inches. 

3i  =  Jj£  X  -V-  =  10.5625  X  3.0317  =  32.022  + 
3  X  3  =  9  X  3.0317  =  27.285  -f- 

Ans.     4.737  X  5  =  23.685  lbs. 

Example.  —  Required  the  weight  of  a  cast  iron  pipe,  whose  length 
is  10  feet,  exterior  diameter  38  i#ches,  and  interior  diameter  3  feet. 
382  X  2.4573  —  362  X  2.4573  =  363.68  X  10  =  3636.8  lbs.     Ans. 


Or,  38*  —  36*  =  148  X  2.4573  =  363.68  X  10  =  3636.8  lbs.     Ans. 

Example.  —  Required  the  weight  of  a  lead  pipe,  whose  length  is 
1200  feet,  exterior  diameter  £  of  an  inch,  and  interior  diameter  A 
of  an  inch. 

I  X  I  =•  *|  =  .765625,  and  A  X  A  =  AV  =  -316406,  and 
.765625  —  .316406  =  .449219  X  3.8697  X  1200  =  2086  lbs.     Ans. 

Example.  —  The  length  of  a  cast-iron  cylinder  is  1  foot,  its 
exterior  diameter  is  12  inches,  and  its  interior  diameter  10  inches ; 
required  its  weight. 

12*  —  10"  =  44  X  2.4573  =  108.12  lbs.     Ans. 
Or,  144  :  353.86  :  :  44  :  108.12  lbs.     Ans. 


WEIGHT   OF    HPES. 


53 


The  following  Table  exhibits  the  coefficients  of  weight,  in  pounds,  of 
one  foot  in  length,  of  various  thicknesses,  of  different  kinds  of  pipe,  of 
any  diameter  whatever. 


Thickness 
in  Inches. 

Wrought 
Iron. 

Copper. 

Lead. 

Jz 

.332 

.379 

.484 

1 
TF 

.664 

.758 

.9675 

A 

.995 

1.137 

1.451 

i 

1.327 

1.516 

1.935 

A 

1.658 

1.894 

2.417 

3 
TF 

1.99 

2.274 

2.901 

A 

2.323 

2.653 

3.386 

J 

2.654 

3.032 

3.87 

A 

3.318 

3.79 

4.837 

1 

3.981 

CAST 

4.548 

IRON. 

5.805 

Thickness. 

Factor. 

Thickness. 

Factor. 

Thickness. 

Factor. 

A 

1.842 

I 

6.143 

»1 

12.287 

i 

2.457 

7.372 

11 

.  14.744 

f 

3.68G 

i 

8.C 

l! 

17.201 

J 

4.901 

i 

9.829 

2 

19.659 

To  obtain  the  weight  of  pipes  by  means  of  the  above  Table  — 
Rule.  —  Multiply  the  diameter  of  the  pipe,  taken  from  the  interior 
surface  of  the  metal  on  the  one  side  to  the  exterior  surface  on  the 
opposite,  (interior  diameter  -f-  thickness,)  in  inches,  by  the  number 
in  the  table  under  the  respective,  metal's  name,  and  opposite  the 
thickness  corresponding  to  that  of  the  pipe  —  the  product  will  be  the 
weight,  in  pounds,  of  one  foot  in  length  of  the  pipe,  and  that  product 
multiplied  by  the  length  of  the  pipe,  in  feet,  will  give  the  weight  for 
any  length  required. 

Example.  —  Required  the  weight  of  a  copper  pipe  whose  length  is 
5  feet,  interior  diameter  and  thickness  3  J  inches,  and  thickness  J  of  an 
inch. 

31  =  fy  =  3.125  X  1-516  X5  =  23.687  lbs.     Ans. 

Example. — Required  the  weight  of  a  cast  iron  pipe,  10  feet  in 
length,  whose  interior  diameter  is  3  feet,  and  whose  thickness  is  1  inch. 
36  -4-  1  =  37  X  9.829  X  10  =  3636.73  lbs.     Ans. 
5* 


54  WEIGHT  OF  BALLS  AND  SHELLS. 

WEIGHT  OF  CAST  IRON  AND  LEAD  BALLS. 

To  find  the  weight  of  a  sphere  or  globe  of  any  material — 
Rule.  —  Multiply  the  cube  of  the  diameter,  in  inches,  or  feet,  by 
the  weight  of  a  spherical  inch  or  foot  of  the  material. 
The  weight  of  a  spherical  inch  of 

Cast  Iron     .   =  .1365  lbs. 
Lead  .         .  =  .215     " 
Therefore  —  Example.  — r  Required  the  weight  of  a  leaden  ball 
whose  diameter  is  £  of  an  inch. 

JX  JX}  =  Vt=  .015625  X  -215  =  .00336  lb.     Ans. 

Example.  —  Required  the  weight  of  a  cast  iron  ball  whose  diameter 
is  8  inches. 

83  X  •  1365  =  69.888  lbs.      Ans. 

Example.  —  How  many  leaden  balls,  having  a  diameter  \  of  an  inch 
each,  are  there  in  a  pound  ? 

1  -7-  .00336  =^U$$m  =  298.     Ans. 

Example.  —  What  must  be  the  diameter  of  a  cast  iron  ball,  to 
weigh  69.888  lbs? 

69.888  -*•  .1365  =  ^/512  =  8  inches.     Ans. 

Example.  — What  must  be  the  diameter  of  a  leaden  ball  to  equal 
in  weight  that  of  a  cast  iron  ball,  whose  diameter  is  8  inches? 
[Lead  is  to  cast  iron  as  .215  to  .1365,  as  1.575  to  1.] 
83  =  512  -j-  1.575  =  ^325  =  6.875  inches.     Ans. 


WEIGHT  OF  HOLLOW  BALLS  OR  SHELLS. 

The  weight  of  a  hollow  ball  is  the  weight  of  a  solid  ball  of  the 
same  diameter,  less  the  weight  of  a  solid  ball  whose  diameter  is  that 
of  the  interior  diameter  of  the  shell. 

Example. — Required  the  weight  of  a  cast  iron  shell  whose  ex- 
terior diameter  is  6|  inches,  and  interior  diameter  4J  inches. 

6\  =2£  X  V-  X  V  =  24414  X  .1365  =  33.33 

4|  =4.253  X  -1365  =  10.48 

22.85  lbs.    Ans. 

Or,  If  we  multiply  the  difference  of  the  cubes,  in  inches,  of  the  two 
diameters  —  the  exterior  and  interior  —  by  the  weight  of  a  spherical 
inch,  we  shall  obtain  the  same  result. 

\mple.  —  Required  the  weight  of  I  cast  iron  shell  whose  ex- 
terior diameter  is  10  inches  and  interior  diameter  8  incht  .->. 
lo  ZZ&  x  .1305  =»  66.612  lbs.     Ans. 


ANALYSIS   OF   COALS. 


55 


ANALYSIS  OF  COALS. 


Description. 

Breckinridge,  Ky., 
"Albert,"  N.  B., 
Chippenville,  Pa. , 
Kanawha,        " 
Pittsburg,        " 
Cannel, 
Newcastle, 
Cumberland, 
Anthracite,  a'v'g. 


Volatile  Matter. 

Carbon. 

62.25 

29.10 

61.74 

32.14 

49.80 

41.85 

32.95 

35.28 

64.72 

24.72 

75.28 

18.40 

80. 

3.43 

89.46 

Ash. 

8.65 
6.12 


1.60 
7.11 


Woods  of  most  descriptions  vary  little  from  80  per  cent,  volatile 
matter,  and  20  per  cent,  charcoal. 

Table  —  Exhibiting  the  Weights,  Evaporative  Powers,  SfC,  of  Fuels, 
.from  Report  of  Professor  Walter  R.  Johnson. 


Weisht 

Lbs.  of  Water 

*t  iM9  degree* 

Lbs.  of  Water 

Weight  of 

Designation  of  Fuel. 

Specific 
Grtv- 

DM 

Cubic 

converted  mio 
Steam  by  I 

at  212  degrees 
converted  into 

Clinkers 
from  100  lbs. 

ity. 

Foot. 

Cubic  Foot  of 

Fuel. 

Steam  by  1  lb. 
of  Fuel. 

of  Coal. 

Anthracite  Coals. 

Beaver  Meadow,  No.  3 

1.610 

54.93 

526.5 

9.21 

1.01 

Beaver  Meadow,  No.  5 

1.554 

56.19 

572.9 

9.88 

.60 

Forest  Improvement 

1.477 

53.66 

577.3 

10.06 

.81 

Lackawanna 

1.421 

48.89 

493.0 

9.79 

1.24 

Lehigh 

1.590 

55.32 

515.4 

8.93 

1.08 

Peach  Mountain 

1.464 

53.79 

581.3 

10.11 

3.03 

Bituminous  Coals. 

Blossburgh 

1.324 

53.05 

522.6 

9.72 

3.40 

Cannclton,  la. 

1.273 

47.65 

360.0 

7.34 

1.64 

Clover  Hill 

1.285 

45.49 

359.3 

7.67 

3.86 

Cumberland,  average, 

1.325 

53.60 

552.8 

10.07 

3.33 

Liverpool 

1.262 

47.88 

411.2 

7.84 

1.86 

Midlothian 

1.294 

54.04 

461.6 

8.29 

8.82 

Newcastle 

1.257 

50.82 

453.9 

8.66 

3.14 

Pictou 

1.318 

49.25 

478.7 

8.41 

6.13 

Pittsburgh 

1.252 

46.81 

384.1 

8.20 

.94 

Scotch 

1.519 

51.09 

369.1 

6.95 

6.63 

Sydney 

1.338 

47.44 

386.1 

7.99 

2.25 

Coke. 

Cumberland 

31.57 

284.0 

8.99 

3.55 

Midlothian 

32.70 

282.5 

8.63 

10.51 

Natural  Virginia 

1.323 

46.64 

407.9 

8.47 

5.31 

Wood. 

Dry  Pine  Wood 

21.01 

98.6 

4.69 

66  MENSURATION   OP   LUMBER. 

MENSURATION  OF  LUMBER. 

To  find  the  contents  of  a  board. 

Rule.  —  Multiply  the  length  in  feet  by  the  width  in  inches,  and 
divide  the  product  by  12  ;  the  quotient  will  be  the  contents  in  square 
feet. 

Example.  —  A  board  is  16  feet  long  and  10  inches  wide;  how 
many  square  feet  does  it  contain  ? 

16  X  10  =  160  -S-  12  —  13^.     Ans. 

To  find  the  contents  of  a  plank,  joist,  or  stick  of  square  timber. 

Rule.  — Multiply  the  product  of  the  depth  and  width  in  inches  by 
the  length  in  feet,  and  divide  the  last  product  by  12  ;  the  quotient  is 
the  contents  in  feet,  board  measure. 

Example.  —  A  joist  is  16  feet  long,  5  inches  wide,  and  2£  inches 
thick ;  how  many  feet  does  it  contain,  board  measure  ? 

5  X  2.5  X  16  -M2  =  16^.     Ans. 

To  find  the  solidity  of  a  plank,  joist,  or  stick  of  square  timber. 

Rule. — Multiply  the  product  of  the  depth  and  width  in  inches 
by  the  length  in  feet,  and  divide  the  last  product  by  144  ;  the  quo- 
tient will  be  the  contents  in  cubic  feet. 

Example.  —  A  stick  of  timber  is  10  by  6  inches,  and  14  feet  in 
length  ;  what  is  its  solidity  1 

10  X  6  =  60  X  14  =*  840  -*■  144  -  5f  feet.     Ans. 

Note.  —  If  aboard,  plank,  or  joist  is  narrower  at  one  end  than  the  other, 
add  the  two  ends  together  and  divide  the  Slim  toy  2j  the  quotient  will  be  the 
mean  width.  And  if  a  stick  of  squared  timber,  whose  solidity  is  required,  is 
narrower  at  one  end  than  the  other  (A  -+-  a  +  S  Aa)  ■+■  3  =  mean  area.  A  and 
a  being  the  areas  of  the  ends. 

To  measure  round  timber. 

Kile  (in  general  practice.^ — Multiply  the  length,  in  feet,  by 
rfie  square  of  \  the  girt,  in  incnes,  taken  about  \  the  distance  from 
the  larger  end,  and  divide  the  product  by  144  ;  the  quotient  is  con- 
sidered the  contents  in  cubic  feet.  For  a  strictly  correct  rule  fix 
measuring  round  timber,  see  Mensuration  of  Solids  —  Frustum  of  a 
Cone. 

Example.  —  A  stick  of  round  timber  is  40  feet  in  length,  and 
girts  88  inches  ;  what  is  its  solidity  1 

88 -i- 4  =  22X22  =  484X40  =  19360  ^-144 -134.44  cub.  ft.  Ans. 


MENSURATION   OF    LUMBER. 


57 


The  following  TABLE  is  intended  to  facilitate  the  measuring  of  Round 
Timber,  and  is  predicated  upon  the  foregoing  Rule. 


i  Girt  in 

Area  in 

i  Girt  in 

Area  in 

i  Girt  in 

Area  in 

i  Girt  in 

Area  in 

Inches. 

Feet. 

Inches. 

Feet. 

Inches. 

Feet. 

Inches. 

Feet. 

6 

.25 

12 

1. 

18 

2.25 

24 

4. 

H 

.272 

12* 

1.042 

184 

2.313 

24* 

4.084 

62 

.294 

124 

1.085 

184 

2.376 

244 

4.168 

61 

.317 

12| 

1.129 

181 

2.442 

241 

4.254 

7 

.34 

13 

1.174 

19 

2.506 

25 

4.34 

n 

.364 

13* 

1.219 

l'.M 

2.574 

25* 

4.428 

n 

.39 

134 

1.265 

194 

2.64 

254 

4.516 

n 

.417 

133 

1.313 

191 

2.709 

251 

4.605 

8 

.444 

14 

1.361 

20 

2.777 

26 

4.694 

H 

.472 

1*4 

1.41 

20* 

2.898 

26* 

4.785 

H 

.501 

144 

1.46 

204 

2.917 

264 

4.876 

8| 

.531 

141 

1.511 

201 

2.99 

261 

4.969 

9 

.562 

15 

1.562 

21 

3.062 

27 

5.062 

H 

.594 

161 

1.615 

21* 

3.136 

27* 

5.158 

H 

.626 

154 

1.668 

214 

3.209 

274 

5.252 

91 

659 

151 

1.722 

211 

3.285 

271 

5.348 

10 

.694 

16 

1.777 

22 

3.362 

28 

5.444 

10* 

.73 

16* 

1.833 

22* 

3.438 

28* 

5.542 

m 

.766 

164 

1.89 

224 

3.516 

284 

5.64 

103 

.803 

161 

1.948 

221 

3.598 

281 

5.74 

11 

.84 

17 

2.006 

23 

3.673 

29 

5.84 

H* 

.878 

17* 

2.066 

23* 

3.754 

29* 

5.941 

114 

.918 

174 

2.126 

234 

3.835 

294 

6.044 

ill 

.959 

171 

2.187 

231 

3.917 

30 

6.25 

To  find  the  solidity  of  a  log  by  help  of  the  preceding  table. 

Rule.  —  Multiply  the  tabular  area  opposite  the  corresponding 
*  girt,  by  the  length  of  the  log  in  feet,  and  the  product  will  be  the 
solidity  in  feet. 

Example.  — The  *  girt  of  a  log  is  22  inches,  and  the  length  of  the 
log  is  40  feet ;  required  the  solidity  of  the  log. 

3.362  X  40  =  134.48  cubic  feet.    Ans. 

Note.  —  Though  custom  has  established,  in  a  very  general  way,  the  preceding  method 
as  that  whereby  to  measure  round  timber,  and  holds,  in  most  instances,  the  solidity  to  be 
that  which  the  method  will  give,  there  seems,  if  the  object  sought  be  the  real  solidity 
of  the  stick,  neither  accuracy,  justice,  nor  certainty,  in  the  practice. 

Thus,  in  the  preceding  example,  the  stick  was  supposed  to  be  40  feet  in  length,  and  88 
inches  in  circumference  at  £  the  distance  from  the  larger  end,  and  was  found,  by  the 
method,  to  contain  134.44  cubic  feet :  now  88  -j-  3.1416  =  28  inches,  =  the  diameter  at  i 
the  distance  from  the  greater  base,  and  retaining  this  diameter  and  the  length,  we  may 


58  MENSURATION   OF   LUMBER. 

suppose,  with  sufficient  liberality,  and  without  being  far  from  the  general  run  of  such 
sticks,  the  diameter  at  the  greater  base  to  be  30  inches,  and  that  of  the  less  to  be  24 
Inches,  and  — 
By  a  correct  rule  the  stick  contains  — 

30  X  24 = 720 -4- 12  =  732  X- 7854X40 =22996-^144  =  159.7  cubic  feet,  or  19  per 
cent,  more  than  given  by  the  method  under  consideration  ;  and  we  need  hardly  add  that 
the  nearer  the  stick  approaches  to  the  figure  of  a  cylinder,  the  wider  will  be  the  difference 
between  the  truth  and  the  result  obtained  by  the  method  referred  to.  Thus,  suppose  tho 
Btick  a  cylinder,  28  inches  in  diameter,  and  40  feet  in  length  ;  and  we  have,  by  the  falla- 
cious rule,  as  above,  134.44  cubic  feet ;  and  — 

By  a  correct  method,  we  have  — 

282X  .7854X40  =  24630 -J- 144 =171  cubic  feet,  or  over  27  per  cent,  more  than  fur- 
nished  by  the  erroneous  mode  of  practice. 

Again  :  suppose  the  stick  in  the  form  of  a  cone,  30  inches  at  the  base,  and  tapering  to  a 
point  at  150  feet  in  length  ;  and  we  have,  by  a  correct  rule  — 

302-r- 3= 300  X -7854X150  =  35343  4-144  =  245.44  cubic  feet;  and  by  the  ordinary 
method  of  gauging,  or  the  aforementioned  practice,  we  have  — 

20  X  3.1416  =  62.832  -7-  4 = 15.7082  X  150  =  37011.19  -7-  144= 257  cubic  feet,  or  nearly 
4}  per  cent,  more  than  the  stick  actually  contains. 

In  short,  without  taking  into  account  anything  for  the  thickness  of  the  bark,  that  may 
be  supposed  to  be  on  the  stick,  the  method  is  correct  only  when  the  stick  tapers  at  the 
rate  of  5i  inches  diameter  per  each  10  feet  in  length,  or  over  i  inch  diameter  to  each  foot 
in  length  of  the  stick. 

If,  however,  we  suppose  the  stick  as  before,  (30  inches  at  the  greater  base,  24  inches  at 
the  smaller,  and  40  feet  in  length,)  and  suppose  the  bark  upon  it  to  be  1  inch  thick,  we 
shall  have,  by  the  usual  method,  134.44  cubic  feet,  as  before.  And,  exclusive  of  the  bark, 
by  a  correct  method,  we  shall  have. 

30  — 2X24  — 2  =  616 -4- 12 =628  X-T854X  40  =  19729 -r  144  =137  cubic  feet,  or 
only  about  2  per  cent,  more  than  that  furnished  us  by  the  usual  practice. 

The  following  simple  rule  for  measuring  round  timber  is  suffi- 
ciently correct  for  most  practical  purposes  :  — 

Rule.  —  Multiply  the  square  of  one-fifth  of  the  mean  girt,  (exclu- 
sive of  bark,)  in  inches,  by  twice  the  length  of  the  stick  in  feet,  and 
divide  the  product  by  144 ;  the  quotient  will  be  the  solidity  in  feet. 

To  find  the  solidity  of  the  greatest  rectangular  stick  that  may  be  cut 
from  a  given  log,  or  from  a  stick  of  round  timber  of  given  dimen- 
sions. 

Rule.  —  Multiply  the  square  of  the  mean  diameter  of  the  log,  in 
inches,  by  half  the  length  of  the  log,  in  feet,  and  divide  the  product 
by  144. 

Example.  — The  diameter  (exclusive  of  bark)  of  the  greater  base 
of  a  stick  of  round  timber  is  30  inches,  and  that  of  the  less  base  is 
24  inches,  and  the  stick  is  40  feet  in  length  ;  required  the  solidity 
of  the  greatest  rectangular  stick  that  may  be  cut  from  it. 

30  X  24  -\-  \  (30  —  24)2  =  732  =  square  of  mean  diameter,*  and 

732  X  20  =  14640 -J- 144  =  101  §  cubic  feet.     Ans.    . 

*  Kxc.ept  in  the  case  of  a  cylinder,  then  is  |  difference  betwixt  the  mr/iti  diameter  of  a 

wing  drcoUr bnei, and  ibtmiddU  dtametar  <>f  thai  Mild.    Dm ne*n dtaaaatar 

reduces  the  solid  to  a  cylinder  ;  the  middle  diameter  is  the  diameter  midway  between  tho 
two 


MENSURATION   Otf   LUMBER.  s  59 

Note.  —  The  foregoing  stick  will  make  — 

14640  — 16  =  915  feet  of  square-edged  boards  1  inch  thick ; 
Or,  101$  ><  9=  91&- 

To  find  the  solidity  of  the  greatest  square  stick  that  may  be  cut  from  a 
given  log,  or  from  a  stick  of  round  timber  of  given  dimensions. 

Rule.  — Multiply  the  square  of  the  diameter  of  the  less  end  of  the 
log,  in  inches,  by  half  the  length  of  the  log,  in  feet,  and  divide 
the  product  by  l44. 

Example.  —  The  preceding  supposed  log  will  make  a  square  stick 
containing  — 

242  X  -42a — 1152  -a- 144— 80  cubic  feet. 

Diameter  multiplied  by  .7071 =  side  of  inscribed  square. 

To  find  the  contents,  in  Board  Measure,  of  a  log,  no  allowance  being 
made  for  wane  or  saw-chip. 

Rule.  — Multiply  the  square  of  the  mean  diameter,  in  inches,  by 
the  length  in  feet,  and  divide  the  product  by  15.28. 

Or,  Multiply  the  square  of  the  mean  diameter  in  inches,  by  the 
length  in  feet,  and  that  product  by  .7854,  and  divide  the  last  prod- 
uct by  12. 

The  cubic  contents  of  a  log  multiplied  by  12,  equal  the  contents 
of  the  log,  board  measure. 

The  convex  surface  of  a  Frustum  of  a  Cone  =  (C  +  c)  X  h  slant 
length ;  C  being  the  circumference  of  the  greater  base,  and  c  the 
circumference  of  the  less. 


60  GAUGING. 

GAUGING. 

Rules  for  finding  the  capacity  in  gallons  or  bushels  of  different 
shaped  Cisterns,  Bins,  Casks,  <5fc,  and  also,  by  way  of  examples,  for 
constructing  them  to  given  capacities. 

Rule  —  1.  When  the  vessel  is  rectangular.  Multiply  the  interior 
length,  breadth,  and  depth,  in  feet  together,  and  the  product  by  the 
capacity  of  a  cubic  foot,  in  gallons  or  bushels,  as  desired  for  its 
capacity. 

Rule  —  2.  When  the  vessel  is  cylindrical.  Multiply  the  square  of 
its  interior  diameter  in  feet,  by  its  interior  depth  in  feet,  and  the  prod- 
uct by  the  capacity  of  a  cylindrical  foot  in  gallons  or  bushels,  as 
desired  for  its  capacity. 

Rule — 3.  When  the  vessel  is  a  rhombus  or  rhomboid.  Multiply 
its  interior  length,  in  feet,  its  right-angular  breath  in  feet,  and  its 
depth  in  feet  together,  and  the  product  by  the  capacity  of  a  cubic  foot 
in  the  special  measure  desired  for  its  capacity. 

Rule —  4.  When  the  vessel  is  a  frustum  of  a  cone — a  round  vessel 
larger  at  one  end  than  the  other,  whose  bases  are  planes.  Multiply 
the  interior  diameter  of  the  two  ends  together,  in  feet,  add  J  the 
square  of  their  difference  in  feet  to  the  product,  multiply  the  sum  by 
the  perpendicular  depth  of  the  vessel  in  feet,  and  that  product  by  the 
capacity  of  a  cylindrical  foot  in  the  unit  of  measure  desired  for  its 
capacity. 

Rule  —  5.  When  the  vessel  is  a  prismoid  or  the  frustum  of  any 
regular  pyramid.  To  the  square  root  of  the  product  of  the  areas  of  its 
ends  in  feet,  add  the  areas  of  its  ends  in  feet,  multiply  the  sum  by 
£  its  perpendicular  depth  in  feet,  and  that  product  by  the  capacity  of 
a  cubic  foot  in  gallons  or  bushels,  as  desired  for  its  capacity. 

If  it  is  found  more  convenient  to  take  the  dimensions  in  inches,  do 
so  ;  proceed  as  directed  for  feet,  divide  the  product  by  1728,  and  mul- 
tiply the  quotient  by  the  capacity  of  the  respective  foot  as  directed. 
Or,  multiply  the  capacity  in  inches  by  the  capacity  of  the  respective 
inch  in  gallons  or  bushels ;  — by  the  quotient  obtained  by  dividing  the 
capacity  of  the  respective  foot  in  gallons  or  bushels  by  1728  —  for 
the  contents. 

Rule  —  6.  When  the  vessel  is  a  barrel,  hogshead,  pipe,  <5fc.  Mul- 
tiply the  difference  in  inches  between  the  bung  diameter  and  head 
diameter,  (interior,)  if  the  staves  be 

much  curved,     .  by  .7  "1 

medium  curved,  .       by  .85  l^g^jwro  go 

straighter  than  medium,  by  .6     [         P  ° 
nearly  straight,  .      by  .55  J 

and  add  the  product  to  the  head  diameter,  taken  in  inches  ;  then  mul 
tiply  the  square  of  the  sum  by  the  length  of  the  cask  in  inches,  and 
divide  the  product  by  the  capacity  in  cylindrical  inches  of  a  gallon  o* 


GAUGING.  61 

bushel  as  desired  for  the  contents.  Or,  divide  the  contents  in  cylin- 
drical inches,  as  above  found,  by  1728,  and  multiply  the  quotient  by 
the  capacity  of  a  cylindrical  foot  in  gallons  or  bushelt  as  desired  for 
its  contents.  Or,  multiply  the  capacity  in  cylindrical  inches  by  the 
capacity  of  a  cylindrical  inch,  in  gallons  or  bushels,  as  desired,  — 
that  is,  by  the  quotient  obtained  by  dividing  the  capacity  of  a  cylin- 
drical foot  in  gallons  or  bushels,  by  1728,  for  the  contents. 
The  capacity  of  a 


CUBIC   FOOT  =3 

7.4805    Winchester  wine  gallons. 

6.1276    Ale 

6.2321    Imperial  " 

.80356  Winchester  bushel. 

.62888  "  heaped    " 

.64285  "  \\  even    " 

.779      Imperial  "      " 


CYLINDRICAL   FOOT  = 

5.8751    Winchester  wine  gallons. 

4.8126    Ale 

4.8947    Impend 
.63111  Winchester  bushel. 

.49391  "  heaped    " 

.50489  "  \\  even    « 

.61183  Imperial  " 


Example. — Required  the  capacity  in  Winchester  bushels  of  a 
rectangular  bin,  whose  interior  length  is  12  feet,  breadth  6  feet,  and 
depth  5  feet. 

12  X  6  X  5  X  -8035  —  289.26  bushels.     Ans. 
Example.  —  Required  the  capacity  in  Winchester  wine  gallons  of 
a  cylindrical  can,  whose  interior  diameter  is  18  inches,  and  depth  3 
feet. 

18  X  18  X  36X  5.875  -f-  1728  =  39.66  gallons.     Ans. 
Or,   1.5  X  1.5  X  3  X  5.875  =*  39.66  gallons.     Ans. 

Or,    18  X  18  X  36  X  .0034  —  39.66  gallons.     Ans. 

Example. — How  many  Winchester  bushels  in  39.66  wine  gal- 
lons? 

39.66  X  .10742  =  4.26  bushels.     Ans. 

Example.  — How  many  wine  gallons  in  4.26  Winchester  bushels  ? 
4.26  X  9.3092  =  39.66  gallons.     Ans. 

Example.  —  How  many  wine  gallons  will  a  cistern  in  the  form  of 
a  frustum  of  a  cone  hold,  having  the  interior  diameter  of  one  of  its 
ends  6  feet,  and  that  at  the  other  8  feet,  and  its  perpendicular  depth 
9  feet? 

8  —  6  =  2,  and  2-  -f-  3  =  1.333  =  £  square  of  dif.  of  diameters,  and 
6X8  +  1.333  =  49.333  X  9  X  5.8751  =  2608.55  gals.     Ans. 

Or,  6  X  8  -f  8'24-62  =  148  X  §  X  5.8751  =  2608.55  gals.  Ans. 
Or,  (83  —  &  )  -^  (8  —  6)  =  148  X  #  X  5.8751  =  2008.55  gals.  Ans. 
Or,  96  —  72  =  24  and  (24s  -r-  3)  =  192,  and 

96  X  72  + 192  =  7104  X108  X  -0034  =  2608.55  gals.     Ans. 
6 


62  GAUGING. 

Example.  —  What  is  the  capacity  in  "Winchester  bushels  of  a  cis- 
tern whose  form  is  prismoid,  the  dimensions  (interior)  of  one  end 
being  8  by  6  feet,  of  the  other  '4  by  3  feet,  and  its  perpendicular 
depth  12  feet? 

8  X  6  =  48  =  area  of  one  end,  and  4  X  3  =  12  =  area  of  the  other 
end  ;  then  — 

48  X  12  =  V576  =  24  +  48  + 12  =  84  XJ#  X  .80356= 270  bush- 
els.    Ans. 

Or,  (8  -|-  4)  -+-  2  =  6,  and  (6  -f-  3)  -J-  2  =  4.5  =  mean  sectional  areas 
of  ends,  and 

6X4. 5X4  =  4  area  of  mean  perimeter,  then 
8  X  6  +  4  X  3  +  6  X  4.5  X  4  =  168  X  "V2- X  .80356  =  270  bus.  Ans. 

Example.  —  What  must  be  the  depth  of  a  rectangular  bin  whose 
length  is  12  feet,  and  breadth  6  feet,  to  hold  289.26  bushels? 
289.26  -T-  (12  X  6  X  .80356)  =  5  feet.     Ans. 

Example.  —  A  cylindrical  can,  whose  depth  is  to  be  36  inches,  is 
required  to  be  made  that  will  hold  40  gallons  ;  what  must  be  the 
diameter  of  the  can  ? 

40  -4-  (3  X  5.8751)  =  V2.27  =  1.506  feet.     Ans. 
Or,  40  ~-  (36  X  .0034)  =  V326.8  =  18.07  inches.     Ans. 

Example.  —  A  cylindrical  can,  whose  interior  diameter  is  to  be  18 
inches,  is  required  that  will  hold  40  gallons  ;  what  must  be  the 
interior  depth  of  the  can  1 

40  -r-  (182  X  .0034)  =  36.31  inches.     Ans. 
Or,  40  -r-  (1.6*  X  5.8751)  =  3.026  feet.     Ans. 

Example.  —  A  cistern  is  to  be  built  in  the  form  of  a  frustum  of  a 
cone,  that  will  hold  1800  gallons,  and  the  diameter  of  one  of  its 
ends  is  to  be  5  feet,  and  that  of  the  other  7£  feet ;  what  must  be  the 
depth ! 

7.5  —  5  —  2.5,  and>2.52  -f-  3  =  2.0833  =  J  square  of  difference  of 
diameter,  and 

1800  -T-  (7.5  X  5  +  2.0833)  X  5.8751  =  7.74  feet.     Ans. 
/7.5X  5  +  7.52  +  52  \ 

Or,  1800  -i-( -J X  5.8751  )=  7.74  feet.     Ans. 

Example.  —  The  form,  capacity,  depth,  and  diameter  of  one  end 
being  determined  on,  and  being  as  above,  what  must  be  the  diameter 
of  the  other  end  ? 

c 
Yj-  —  \d2  =  y,c  being  the  solidity  in  cylindrical  measurement,  h 


GAUGING.  63 

the  depth,  d  the  diameter  of  the  given  end  or  base,  and  y  a  quantity 
the  square  root  of  which  is  the  sum  of  the  required  base  and  half  the 
given  base  ;  then 

1800  -T-  5.8751  =  306.378  =  solidity  in  cylindrical  feet,  and 

306.378  -j-  fcf*  =  118.75  —  (52  -±  %  )  =»  V100  -  10  —  £  =  7.5 
feet.     Ans. 

Example.  —  A  measure  is  to  be  built  in  the  form  of  a  frustum  of  a 
cone,  that  will  hold  exactly  1  wine  gallon,  and  the  diameter  of 
one  of  its  ends  is  to  be  4  inches,  and  that  of  the  other  6  inches ; 
what  must  be  its  depth  1 

1  -T-  (6  X  4  +  U)  X  -0034  =  11.61  inches.     Ans. 

231         6  X  4  -f  62  -f  42 
Or,    nQZA  -5- 3 =  11.61  inches.   Ans. 

Example.  —  A  measure  in  the  form  of  a  frustum  of  a  cone  holds  1 
wine  gallon  ;  the  diameter  of  one  of  its  ends  is  6  inches,  and  its 
depth  is  11.61  inches  ;  what  is  the  diameter  of  the  other  end  ? 

Jilt  =  294.1176  h-  Llir6-1-  =  76  -  (6*  -J-  | )  =  V49  -  7  —  } 
=  4  inches.     Ans. 


CASK   GAUGING. 

Cask-gauging,  in  a  general  sense,  is  a  practical  art,  rather 
than  a  scientific  achievement  or  problem,  and  makes  no  pretensions 
to  strict  accuracy  with  regard  to  the  conclusions  arrived  at.  The 
aim  is,  by  means  of  a  few  satisfactory  measurements  taken  of  the 
outside,  and  an  estimate  of  the  probable  mean  thickness  of  the  ma- 
terial of  which  the  cask  is  composed  (of  which  there  must  always 
remain  some  doubt),  or  by  means  of  a  few  measurements  taken  of , 
the  inside,  to  determine,  1st,  the  capacity  of  the  cask,  and,  2d,  the 
ullage,  or  capacity  of  the  occupied  or  unoccupied  space  in  a  cask 
but  partly  full.  And  the  Ruie  (Rule  6,  page  60^),  which  re- 
duces the  supposed  cask,  or  cask  of  supposed  curvature,  to  a  cylin- 
der, is  as  practically  correct  for  the  capacity  of  ordinary  casks,  as 
any  rule,  or  set  of  rules,  that  can  be  offered  for  general  purposes. 

Casks  have  no  fixed  form  of  their  own,  to  which  they  severally 
and  collectively  correspond,  nor  are  they  in  any  considerable  degree 
in  conformity  with  any  regular  geometrical  figure. 

Some  casks  —  a  few  —  those  having  their  staves  much  curved 
throughout  their  entire  length,  are  nearest  in  keeping  with  the 
middle  frustum  of  a  spheroid ;  others,  slightly  less  curved  than 
the  preceding,  correspond  in  a  considerable  degree  to  the  middle 


64  GAUGING. 

frustum  of  a  parabolic  spindle;  others,  again  —  thoso  having  very 
little  longitudinal  curvature  of  stave  to  their  semi-lengths  —  are 
nearly  in  keeping  with  the  equal  frustums  of  a  paraboloid ;  and  others 
—  a  very  few  —  those  whose  staves  are  straight  from  the  bung  diam- 
eter to  the  heads,  or  equal  to  that  form,  are  in  accordance  with  the 
equal  frustums  of  a  cone. 

The  gauging  rod,  which  is  intended  to  be  correct  for  casks  of  the 
most  common  form,  gives  for  all  casks,  as  may  be  seen  in  one  of  the 
following  Examples,  a  solidity  slightly  greater  (about  2£  per  cent.) 
than  would  be  obtained  by  supposing  the  cask  in  conformity  with 
the  third  figure  above  alluded  to. 

The  Rule  for  finding  the  contents  of  a  cask,  by  four  dimensions, 
hereafter  to  be  given,  is  intended  as  a  general  Rule  for  all  casks, 
and,  when  the  diameter  midway  between  the  bung  and  head  can  be 
accurately  ascertained,  will  lead  to  a  very  close  approach  to  the 
truth. 

From  the  length  of  a  cask,  taken  from  outside  to  outside  of  the 
heads,  with  callipers,  it  is  usual  to  deduct  from  1  to  2  inches,  to  cor- 
respond with  the  thickness  of  the  heads,  according  to  the  size  of  the 
cask,  and  the  remainder  is  taken  as  the  length  of  the  interior. 

To  the  diameter  of  each  head,  taken  externally,  from  \  inch  to 
y'V  inch  should  be  added  for  common-sized  barrels,  -£$  inch  for  40  gal- 
lon casks,  and  from  £  inch  to  ^  inch  for  larger  casks,  to  correspond 
with  the  interior  diameters  of  the  heads. 

If  the  staves  are  of  uniform  thickness,  any  sectional  diameter  of  a 
cask  may  be  nearly  or  quite  ascertained,  by  dividing  the  circumfer- 
ence at  that  place  by  3.141G,  and  subtracting  twice  the  thickness  of 
the  stave  from  the  quotient. 

For  obtaining  the  diagonal  of  a  cask  by  mathematical  process,— 
the  interior  length,  &c.  &c.  — see  Rules,  below. 

In  the  following  formulas  D  denotes  the  bung  diameter,  d  the 
head  diameter,  and  /  the  length  of  the  cask. 

The  solidity  of  any  cask  is  equal  to  its  length  multiplied  by  the 
square  of  its  mean  diameter  multiplied  by  .7854. 

To  calculate  the  contents  of  a  cask  from  four  dimensions. 

Rule.  —  To  the  square  of  the  bung  diameter  add  the  square  of 
the  head  diameter,  and  the  square  of  double  the  diameter  midway 
between  the  bung  and  head,  and  multiply  the  sum  by  £  the  length 
of  the  cask,  for  its  cylindrical  contents;  the  product  multiplied  by 
.Oil  ;  I  expresses  the  contents  in  wine  gallons. 

Bl ami'LK.  — The  length  of  the  cask  is  40  inchos,  its  bung  diameter 
28  inches,  head  diameter  20  inches,  and  the  diameter  midway  be- 


QAUGINQ.  65 

tween  the  bung  and  head  is  25. G  inches ;  how  many  gallons'  capacity 
has  the  cask  1  ' 

202 -f  28-'  +  25.6  X  2*  =  3805.44  X  V"  X  .0034  =  86.26  gals.     Am. 

(D2  -f-  <F  -f  2m  )X^X  -7854  =  cubic  contents. 

- =  square  of  mean  diameter. 

By  Rule  6,  p.  68,  this  cask  will  hold  — 

28  —  20  =  8  X  .65  =  5.2  -|-  20  =  25.2  X  25.2  X  40  X  .0034  —  86.36 
gallons. 

When  the  cask  is  in  the  form  of  the  middle  frustum  of  a  spheroid. 

|  D2-{-  J  d 2  =  square  of  mean  diameter. 

And  a  cask  of  this  form,  having  the  same  head  diameter,  bung 
diameter,  and  length  as  the  preceding,  will  hold  — 

2X282  +  2Q2X40  X  .0034  =* 89.216  gallons, 
o 

When  the  cask  is  in  the  form  of  the  middle  frustum  of  a  parabolic 
spindle. 

f  D2-]-  J  d'2  —  j25  (D^  dy  as  square  of  mean  diameter. 
And  a  cask  of  this  form,  having  the  same  head  diameter,  bung 
diameter,  and  length  as  the  preceding,  will  hold  — 

522§  + 133|=  656  —  8.533  =  647.467  X  40  X  .0034  s  88.055  gals. 

When  the  cask  is  in  the  form  of  two  equal  frustums  of  a  paraboloid, 
h  D2-j-  h  d  2  =  square  of  mean  diameter. 

And  a  cask  of  this  form,  having  the  same  head  diameter,  bung 
diameter,  and  length  as  the  preceding,  will  hold  — 

28«  +  202 

f- X  40  X  -W34  =s  80.51  gallons. 

When  the  cask  is  in  the  form  of  the  equal  frustums  of  a  cone. 
£D2_j_£  d~  —  i  (D  ^r  d)2  ==  square  of  mean  diameter. 
Or,  ^D2+^+JDrf==     "       "      " 
Or,  DX^-H(I>^rf)2=  "       "      " 
And  a  cask  of  this  form,  having  the  same  head  diameter,  bung 
diameter,  and  length  as  the  preceding,  will  hold  — 

28  X  20  +  21J  X40X-0034  =  79.06  gals. 
6* 


66  GAUGING. 

To  find  the  contents  of  a  cask  the  same  as  would  be  given  by  the 
gauging  rod. 

The  gauging  rod  is  constructed  upon  the  principle  that  the  oabe 
of  the  diagonal  of  a  cask,  in  inches,  multiplied  by  ^j^jrj>  equals  the 
contents  of  the  cask,  in  Imperial  gallons. 

The  contents  in  wine  gallons  of  either  of  the  aforementioned 
casks,  therefore,  by  the  gauging  rod,  would  be  — 

3T24F  X  .0027  «  82J  gals. 

The  decimal  coefficient  to  take  the  place  of  .0027,  for  finding  the 
contents  of  a  cask  in  the  form  of  the  middle  frustum  of  a  spheroid 
=  .002926  ;  and  for  finding  the  contents  of  a  cask  in  the  form  of  the 
equal  frustums  of  a  cone  =  .002593.  And  between  these  extremes 
lies  the  decimal  for  other  casks,  or  casks  of  intervening  figures. 

To  find  the  diagonal  of  a  cask,  when  the  interior  is  inaccessible. 

Rule.  —  From  the  bung  diameter  subtract  half  the  difference  of 
the  bung  and  head  diameters,  and  to  the  square  of  the  remainder 
add  the  square  of  half  the  length  of  the  cask,  and  the  square  root 
of  the  sum  will  be  the  diagonal. 

Example.  —  What  is  the  diagonal  of  a  cask  whose  bung  diameter 
is  28  inches,  head  diameter  20  inches,  and  length  40  inches? 

28-20  =  8-h2  =  4,and28  — 4  =  24,  then 

V  (242  _j_  20-')  =*  31.241  inches.     Ans. 

To  find  the  length  of  a  cash,  the  head  diameter,  bung  diameter  and 
diagonal  being  given. 

V I  diagonal2  —  D  —  — ^ — )  =  £  /. 

And  the  interior  length  of  a  cask,  whose  interior  head  diameter, 
bung  diameter  and  diagonal,  are  as  the  preceding,  will  be 
V  (31.2412  —  242)  =  20  X  2  =  40  incne8. 

To  find  the  solidity  of  a  sphere. 
D2  X  I  D  X  .7854  =  cubic  contents,  D  being  the  diameter. 

To  find  the  solidity  of  a  spherical  frustum. 

I  3  A2  +  — ^ —  )XhX  .7854 » cubic  contents, b  and  d being tho 

bases,  and  h  the  height. 

None.  —  Hr  Kul' s  in  di-uil  iK.TUiiiiing  to  the  foregoing  figures,  and  for  ether  figures, 
•ee  Mensuration  or  Souns. 


ULLAGE.  67 

ULLAGE. 

The  ullage  or  wantage  of  a  cask  is  the  quantity  the  cask  lacks  of 
being  full. 

To  find  the  ullage  of  a  standing  cask,  when  the  cask  is  half  f  nil  or  more. 

Rule. — To  the  square  of  the  head  diameter,  add  the  square  of 
the  diameter  at  the  surface  of  the  liquor,  and  the  square  of  twice 
the  diameter  midway  between  the  surface  of  the  liquor  and  the  upper 
head,  and  divide  the  sum  by  6 ;  the  quotient,  multiplied  by  the 
distance  from  the  surface  of  the  liquor  to  the  upper  head,  multiplied 
by  .0034,  will  give  the  ullage  in  wine  gallons. 

Example.  — The  diameters  are  as  follows  —  at  the  upper  head,  20 
inches  ;  at  the  surface  of  the  liquor,  22  inches  ;  and  at  a  point  midway 
between  these,  21^  inches ;  and  the  distance  from  the  upper  head 
to  the  surface  of  the  liquor  is  5  inches ;  required  the  ullage. 

(20*  -f  22*+  21.25  X  22)  -j-  6  =  448.37  X  5  X  .0034  =  7.62  gal- 
Ions.     Ans, 

When  the  cask  is  standing,  and  less  than  half  full,  to  find  the  ullage. 

Rule.  —  Make  use  of  the  bung  diameter  in  place  of  the  head 
diameter,  and  proceed  in  all  respects  as  directed  in  the  last  Rule, 
and  add  the  quantity  found  to  half  the  capacity  of  the  cask ;  the 
sum  will  be  the  ullage.  j 

Example.  —  The  bung  diameter  is  28  inches ;  the  diameter  at  the 
surface  of  the  liquor,  below  the  bung,  is  26  inches ;  the  diameter 
midway  between  the  bung  and  the  surface  of  the  liquor  is  27.3 
inches  ;  and  the  distance  from  the  surface  of  the  liquor  to  the  bung 
diameter  is  5  inches ;  required  the  quantity  the  cask  lacks  of  being 
half  full ;  and  also  the  ullage  of  the  cask,  its  capacity  being  86.26 
gallons. 

(282  -f-26*  -t-  27.3X2*)  H-  6  =  740.2  X  5  X  -0034  =  12.58  gal-' 

Ions  less  than  £  full.     Ans, 
And,  86.2G  -j-  2  =  43.13  -f- 12.58  =  55.73  gallons  ullage.     Ans. 

When  the  cask  is  upon  its  bilge,  and  half  full  or  more,  to  find  the  ullage. 

Rule.  —  Divide  the  distance  from  the  bung  to  the  surface  of  the 
liquor  —  (the  height  of  the  empty  segment)  —  by  the  whole  bung 
diameter,  and  take  the  quotient  as  the  height  of  the  segment  of  a 
circle  whose  diameter  is  1,  and  find  the  area  of  the  segment;  mul- 
tiply the  area  by  the  capacity  of  the  cask,  in  gallons,  and  that 
product  by  1.25  ;  the  last  product  will  be  the  ullage,  in  gallons,  as 


68  ULLAGE. 

found  by  the  aid  of  the  wantage-rod ;  and  will  be  correct  for  casks 
of  the  most  common  form. 

Note.  —  The  area  of  the  segment  of  a  circle  = 

(ch'd  i  arc  -j-  4  ch'd  i  arc  -|-  ch'd  seg.)  X  height  seg.  X  y4o"*»  yer*  nearly> 
And,  having  the  diameter  of  the  circle  and  the  height  of  the  segment  given,  the  chord 
of  half  the  arc,  and  the  chord  of  the  segment  may  be  found,  thus  — 
radius  —  height  =  cosine ;  radius2  —  cosine2  =  sine2 ;  */ {sine  )  X  2  =  ch'd  of  seg. 
sine2  +  heigkt  seg.2  =  ch'd  i  arc2,  and  +/  (ch'd  i  arc2)  =  ch'd  *  arc. 

Example. — The  bung  diameter  is  28  inches,  the  height  of  the 
empty  segment  5.6  inches,  and  the  capacity  of  the  cask  86.26  gal- 
lons ;  required  the  ullage  of  the  cask,  in  gallons. 

5.6  -r  28  =  .2  =  height  of  seg.,  diameter  as  1. 
1  -r-  2  =  .5  =  radius. 
.5  —  .2  —  .3  =  cosine. 

.52  —  .32  =  .16  =  sine2,  or  square  of  half  the  base  of  the  segment. 
V-16"  =  .4  X  2  =  .8  =  chord  of  segment,  or  base  of  segment. 
.4?  -{-  .22  =  .2  =  square  of  chord  of  half  the  arc. 
V-2  =  .4472  =  chord  of  half  the  arc,  then  — 
.4472  -j-  3  =  .1491,  and  .1491  +  .4472  -f-  .8  X  -2  X  A  =  -1H7, 
area  of  segment,  and 

.1117  X  86.26  X  1.25  =  12  gallons.     Ans. 

When  the  cask  is  upon  its  bilge,  and  less  than  half  full,  to  find  the 

ullage. 
Rule.  —  Divide  the  depth  of  the  liquor  by  the  bung  diameter,  and 
proceed  in  all  respects  as  directed  in  the  last  Rule  ;  then  subtract 
the  quantity  found  from  the  capacity  of  the  cask,  and  the  difference 
will  be  the  ullage  of  the  cask. 

To  find  the  quantity  of  liquor  in  a  cask  by  its  weight. 

Example.  — The  weight  of  a  cask  of  proof  spirits  is  300  lbs.,  and 
the  weight  of  the  empty  cask  {tare)  is  32  lbs.  How  many  gallons 
are  there  of  the  liquor  ? 

300  —  32  =  268  ~  7.732  =  345  gallons.    Ans. 

Customary  Rule  by  Freighting  Merchants,  for  finding  the  cubic  meas- 
urement of  casks. 

Bung  diameter2  X  £  length  of  cask  =  cubic  measurement. 

Note.  —  One  cubic  foot  contains  7.4805  wine  gallons. 

*  For  several  Rules  in  detail,  for  finding  the  area  of  the  segment  of  a  circle,  sec  Geom- 
*m— Mensuration  of  Superficies. 


TONNAGE. 

TONNAGE. 

GOVERNMENT   MEASUREMENT. 


length  —  f  breadth  X  breadth  X  depth 
^ =  tonnage. 

In  a  double-decked  vessel,  the  length  is  reckoned  from  the  fore 
part  of  the  main  stem  to  the  after  side  of  the  sternpost  above  the 
upper  deck ;  the  breadth  is  taken  at  the  broadest  part  above  the 
main  wales,  and  half  this  breadth  is  taken  for  the  depth. 

In  a  single-decked  vessel  the  length  and  breadth  are  taken  as  for 
a  double-decked  vessel,  and  the  distance  between  the  ceiling  of  the 
hold  and  the  under  side  of  the  deck  plank  is  taken  as  the  depth. 

Example.  —  The  length  of  a  double-decked  vessel  is  260  feet,  and 
the  breadth  is  60  feet ;  required  the  tonnage. 

260  —  -MP-  =  224  X  60  X  -6#-  =  403200  -7-  95  =  4244.2  tons.  Ans. 

Example.  —  The  length  of  a  single-decked  vessel  is  180  feet,  the 
breadth  34  feet,  and  depth  18  feet ;  required  the  tonnage. 

180  —  f  of  34  =  159.6  X  34  X  18  -*•  95  —  1028.16  tons.    Ans. 

CARPENTER'S  MEASUREMENT. 

For  a  double-decked  — 

length  of  keel  X  breadth  main  beam  X  h  breadth 
95 =  tonnage. 

For  a  single-decked  — 
length  of  keel  X  breadth  main  beam  X  depth  of  hold 


95 


■= tonnage, 


70  CONDUITS  OK   PIPES, 


OF  CONDUITS  OR  PIPES. 
Pressure  of  Water  in  Vertical  Pipes,  <5fC. 

h  =  height  of  column  in  inches  ;  o  =  circumference  of  column  in  inches-; 
t  =  thickness  of  pipe  in  inches  equal  in  strength  to  lateral  pressure  at  bas« 
of  column  ;  w  am  weight  of  a  cubic  inch  of  water  in  pounds  ;  C  =  cohesive 
strength  in  pounds  per  inch  area  of  transverse  section  of  the  material  of 
which  the  pipe  is  composed  —  table,  p.  72. 

h  o  =  area  of  interior  of  pipe  in  inches  ;  h  w  =  pressure  in  pounds  per 
square  inch  at  the  base  of  the  column,  or  maximum  lateral  pressure  in 
pounds  per  square  inch  on  the  pipe  tending  to  burst  it  ;  how  =  maximum 
lateral  pressure  in  pounds  on  the  pipej  tending  to  burst  it  at  the  bottom  -r 
and  how  ■—•  2  =  mean  lateral  pressure  in  pounds  on  the  pipe,  or  pressure 
in  pounds  on  the  pipe  tending  to  burst  it  at  half  the  height  of  the  column. 

how-r-C  =  t;  how-i-t=C;  Ct-r-ow  =  h;  Qt  -—  hw  =  o. 

Notb.  —The  reliable  cohesion  of  a  material  is  not  above  4  its  ultimate  force,  as  given 
in  the  Table  of  Cohesive  Forces.  By  experiment,  it  has  been  found  that  a  cast  iron  pipe 
15  inches  in  diameter  and  $  of  an  inch  thick,  will  support  a  head  of  water  of  600  feet ;  and 
that  one  of  the  same  diameter  made  of  oak,  and  two  inches  thick,  will  support  a  head  of 
ISO  feet :  12000  lbs.  per  square  inch  for  cast  rron,  1200  for  oak,  750  for  lead,  are  counted 
safe  estimates.  The  ultimate  cohesion  of  an  alloy,  composed  of  lead  8  parts  and  zinc  I 
part,  is  3000  pounds  per  square  inch. 

Concerning  the  Discharge  of  Pipes,  <3{C. 

Small  pipes,  whether  vertical,  horizontal,  or  inclined,  under  equal 
heads,  discharge  proportionally  less  water  than  large  ones.  That 
form  of  pipe,  therefore,  which  presents  the  least  perimeter  to  its  area, 
other  things  being  equal,  will  give  the  greatest  discharge.  A  round 
pipe,  consequently,  will  discharge  more  water  in  a  given  time  than  a 
pipe  of  any  other  form,  of  equal  area. 

The  greater  the  length  of  a  pipe  discharging  vertically,  the  greater 
the  discharge.  Because  the  friction  of  the  particles  against  its  sides, 
and  consequent  retardation,  is  more  than  overcome  by  the  gravity  of 
the  fluid. 

The  greater  the  length  of  &  pipe  discharging  horizontally,  the  less 
proportionally  will  be  the  discharge.  The  proportion  compared  with 
a  less  length  is  in  the  inverse  ratio  of  the  square  root  of  the  two 
lengths,  nearly. 

Other  things  being  equal,  rectilinear  pipes  give  a  greater  discharge 
than  curvilinear,  and  curvilinear  greater  than  angular.  The  head, 
the  diameters  and  the  lengths  being  the  same,  the  time  occupied  in 
passing  an  equal  quantity  of  water  through  a  straight  pipe  is  9, 
through  one  curved  to  a  semicircle  10,  and  through  one  having  one 
right  aiiijle,  otherwise  straight,  11.  All  interior  inequalities  ;in<l 
roughness  should  be  avoided. 

It  has  been  ascertained  that  a  velocity  of  60  feet  a  minute  (1  foot 
a  second)  through  a  horizontal  pipe,  4  inches  in  diameter  and'  100  feet 


CONDUITS   OR   PIPES.  71 

in  length,  is  produced  by  a  head  2}  inches,  only  f  of  an  inch  above 
the  upper  surface  of  the  orifice  ;  and  that,  to  maintain  an  equal 
velocity  through  a  pipe  similarly  situated,  of  equal  length,  having  a 
diameter  of  \  inch  only,  a  head  of  1  fa  feet  *s  required.  To  increase 
the  velocity  through  the  last  mentioned  pipe  to  2  feet  a  second, 
requires  a  head  4|£  feet ;  to  3  feet,  a  head  of  10T^- ;  to  4  feet,  a 
head  of  17|^,  &c. 

From  the  foregoing,  the  following,  it  is  believed,  reliable  rules,  are 
deduced. 

To  find  the  velocity  of  water  passing  through  a  straiglit  horizontal  pipe 
of  any  length  and  diameter,  the  head,  or  height  of  the  fluid  above  the 
centre  of  the  orifice,  being  known. 

Rule.  —  Multiply  the  head,  in  feet,  by  2500,  and  divide  the  product 
by  the  length  of  the  pipe,  in  feet,  multiplied  by  13.9,  divided  by  the 
interior  diameter  of  the  pipe  in  inches  ;  the  square  root  of  the  quotient 
will  be  the  velocity  in  feet  per  second. 

Example.  —  The  head  is  6  feet,  the  length  of  the  pipe  1340  feet, 
and  its  diameter  5  inches  ;  required  the  velocity  of  the  water  passing 
through  it. 

2500  X  6  ==  15000  -7-  (IfliOff  13-9)  -*-  ^4.03  —  =  2  feet  per 
second.     Ans. 

To  find  the  head  necessary  to  produce  a  required  velocity  through  a  pipe 
of  given  length  and  diameter. 
Rule.  —  Multiply  the  square  of  the  required  velocity,  in  feet,  per 
second,  by  the  length  of  the  pipe  multiplied  by  the  quotient  obtained 
by  dividing  13.9  by  the  diameter  of  the  pipe  in  inches,  and  divide  the 
product  thus  obtained  by  2500  ;  the  quotient  will  be  the  head  in  feet. 

Example. — The  length  of  a  pipe  lying  horizontal  and  straight   is 
1340  feet,  and  its  diameter  is  5  inches  ;  what  head  is  necessary  to 
cause  the  water  to  flow  through  it  at  the  rate  of  2  feet  a  second  ? 
2'2  X  1340  X  H~  -T-  2500  =  6  feet.     Ans. 

To  find  the  quantity  of  water  flowing  through  a  pipe  of  any  length  and 
diameter. 
Rule.  —  Multiply  the  velocity  in  feet  per  second  by  the  area  of  the 
discharging  orifice,  in  feet,  and  the  product  is  the  quantity  in  cubic 
feet  discharged  per  second. 

Example.  —  The  velocity  is  2  feet  a  second,  and  the  diameter  of 
the  pipe  5  inches;  what  quantity  of  water  is  discharged  in  each 
second  of  time  ? 
5  -f-  12*=  .4166,  and  .41662  X  -7854  X  2  =  .273  cubic  foot.     Ans. 


72  MISCELLANEOUS    FHOBLEMS. 

i 

MISCELLANEOUS  PROBLEMS. 
To  find  the  specific  gravity  of  a  body  heavier  than  water. 

Rule.  —  Weigh  the  body  in  water  and  out  of  water,  and  divide  the 
weight  out  of  water  by  the  difference  of  the  two  weights. 

Example.  —  A  piece  of  metal  weighs  10  lbs.  in  atmosphere,  and 
but  8i  in  water ;  required  its  specific  gravity. 

10  —  8.25  —  1.75,  and  10  -f-  1.75  =  5.714.     Ans. 

To  find  the  specific  gravity  of  a  body  lighter  than  water. 
Rule.  —  Weigh  the  body  in  air ;  then  connect  it  with  a  piece  of 
metal  whose  weight,  both  in  and  out  of  water,  is  known,  and  of  suf- 
ficient weight  that  the  two  will  sink  in  water  ;  and  find  their  combined 
weight  in  water ;  then  divide  the  weight  of  the  body  in  air  by  the 
weight  of  the  two  substances  in  air,  less  the  sum  of  the  difference  of 
the  weight  of  the  metal  in  air  and  water  and  the  combined  weight  of 
the  two  substances  in  water,  and  the  quotient  will  be  the  specific 
gravity  sought. 

Example.  —  The  combined  weight,  in  water,  of  a  piece  of  wood, 
and  piece  of  metal,  is  4  lbs. ;  the  wood  weighs  in  atmosphere  10  lbs. ; 
and  the  metal  in  atmosphere  12,  and  in  water  11  lbs. ;  required  the 
specific  gravity  of  the  wood. 

10  -r-  (10  -{-  12  —  12  ^  11  -f  4)  =  .588.     Ans. 
To  find  the  specific  gravity  of  a  fluid. 
Rule.  — Multiply  the  known  specific  gravity  of  a  body  by  the  dif- 
ference of  its  weight  in  and  out  of  the  fluid,  and  divide  the  product  by 
its  weight  out  of  the  fluid  ;  the  quotient  will  be  the  specific  gravity  of 
the  fluid  in  which  the  body  is  weighed. 

Example.  — The  specific  gravity  of  a  brass  ball  is  8.6  ;  its  weight 
in  atmosphere  is  8  oz.,  and  in  a  certain  fluid  7{  oz. ;  required  the 
specific  gravity  of  the  fluid. 
8  _  7.25  =  .75,  and  8.6  X  -75  =  6.45,  and  6.45  -H  8  =  .806.    Ans. 

To  find  the  proportion  of  one  to  the  other  of  two  simples  forming  a 
compound,  or  the  extent  to  which  a  metal  is  debased,  (the  metal  and  the 
alloy  used  being  knoun.) 

The  Rule  strictly  bears  upon  that  of  Alligation  Alternate,  which 
see. 

Example. — The  specific  gravity  of  gold  is  19.258,  and  that  of 
copper,  8.788;  an  article  composed  of  the  two  metals,  has  a  specific 
gravity  of  18  ;  in  what  proportion  are  the  metals  mixed] 
18^  19.258  X    8.788=11.055 
18  ^     8.788  X  19-258  =    177.4,  then 


MISCELLANEOUS    PROBLEMS.  73 


11.055  +  177.4  :   11.055  :  :  18  —  1.056  oopper,  )  An$ 
11.055  4-177.4  :     177.4  :  :   18  «  16.944  gold.    S 
Or,  18  —  1.056  =  16.914  gold.     Copper  to  gold  as  1  to  16.04  + 

To  find  the  lifting  fower  of*  balloon. 
Rule.  — Multiply  the  capacity  of  the  balloon,  in  feet,  by  the  dif- 
ference of  weight  between  a  cubic  foot  of  atmosphere  and  a  cubic  foot 
of  the  gas  used  to  inflate  the  balloon,  and  the  product  is  the  weight 
the  balloon  will  raise. 

Example.  — A  balloon,  whose  diameter  is  24  feet,  is  inflated  with 
hydrogen  ;  what  weight  will  it  raise  1 

Specific  gravity  of  air  is  1,  weight  of  a  cubic  foot  527.04  grains; 
specific  gravity  of  hydrogen  is  .0689. 

527.04  X  '0689  =  8(5,31  grains  =  weight  of  1  cubic  foot  of  hydrogen. 
527.04  —  36.31  =  490.73  grs.  =  dif.  of  weight  of  air  and  hydrogen. 
243  X  .5236  =  7238.24  =  capacity  in  cubic  feet  of  balloon. 
Then,   7238.24  X  490.73  =  35,52021  grs.  =  i^$>£-L  =  507TV  lbs. 

Ans. 

To  find  the  diameter  of  «  balloon  that  shall  be  equal  to  the  raising  of  a 

given  weight. 

The  weight  to  be  raised  is  507^  lbs. 

507T4X  7000-7-490.73  =  7238.24,  and  7238.24  -f-  .5236  =  fy  13824 

=  24  feet.     Ans. 

To  find  the  thickness  of  a  concave  or  Jiollow  metallic  ball  or  globe,  tfiat  shall 
have  a  given  buoyancy  in  a  given  liquid. 

Example.  —  A  concave  globe  is  to  be  made  of  brass,  specific  grav- 
ity 8.6,  and  its  diameter  is  to  be  12  inches;  what  must  be  its  thick- 
ness that  it  may  sink  exactly  to  its  centre  in  pure  water  T 

Weight  of  a  cubic  inch  of  water  .036169  lb. ;  of  the  brass  .3112  lb. 
Then,  123  X  -5236  X  .036169  -r-  2  =  16.3625  cubic  inches  of  water 
to  be  displaced.  , 

16.3625  -=-  .3112  =  52.5787  cubic  inches  of  metal  in  the  ball. 

122  X  3.1416  =  452.39  square  inches  of  surface  of  the  ball. 
And,  52.5787  -f-  452.39  =  .1162  +  =  £  inch  thick,  full.     Ans. 

To  cut  a  square  sheet  of  copper,  tin,  etc.,  so  as  to  form  a  vessel  of  the 
greatest  cubical  capacity  the  sheet  admits  of. 

Rule.  —  From  each  corner  of  the  sheet,  at  right  angles  to  the  side, 
cut  £  part  of  the  length  of  the  side,  and  turn  up  the  sides  till  the 
corners  meet. 

7 


74 


COMPARATIVE    COHESIVE    FORCE. 


Comparative  Cohesive  Force  of  Metals,  Woods,  and  other  substances, 
Wrought  Iron  (medium  quality)  being  the  unit  of  comparison,  or  1  ; 
the  cohesive  force  of  which  is  60000  lbs.  per  inch,  transverse  area. 


Wrought  iron,  . 

.     1.00 

Ash,  white, 

.23 

"         "     wire,  . 

1.71 

"     red, 

.30 

Copper,  cast,    . 

,       .40 

Beech,      . 

.19 

"       wire,   . 

.76 

Birch, 

.25 

Gold,  cast, 

.34 

Box,         . 

.33 

"     wire, 

.51 

Cedar,      . 

.19 

Iron,  cast,  (average). 

.       .38 

Chestnut,  sweet, 

.17 

Lead,    " 

.015 

Cypress,  . 

.10 

"     milled,     . 

.     .055 

Elm,         .         .       '  . 

.22 

Platinum,  wire, 

.88 

Locust,     . 

-34 

Silver,  cast, 

.66 

Mahogany,  best, 

36 

"      wire,     . 

.68 

Maple, 

.18 

Steel,  soft,    *     . 

2.00 

Oak,  Amer.,  white,  . 

.19 

"     fine, 

2.25 

Pine,  pitch, 

.20 

Tin,  cast  block, 

.083 

Sycamore, 

.22 

Zinc,  "              . 

.043 

Walnut,    . 

.30 

"     sheet, 

.27 

Willow,   . 

.22 

Brass,  cast, 

.75 

Ivory, 

.27 

Gun  metal, 

.50 

Whalebone,     . . 

.13 

Gold  5,  copper  1, 

.       .83 

Marble,    . 

.15 

Silver  5,     "       1,      . 

.80 

Glass,  plate, 

.16 

Brick, 

.05 

Hemp  fibres,  glued,  . 

.     1.53 

Slate, 

.20 

The  strength  of  white  oak  to  cast  iron,  is  as  2  to    9. 
The  stiffness  of      "      "         "         "     is  as  1  to  13. 

To  determine  the  vmght,  or  force,  in  pounds,  necessary  to  tear  asun- 
der a  bar,  rod,  or  piece  of  any  of  the  above  named  substances,  of  any 
given  transverse  area : 

Rule.  —  Multiply  the  comparative  cohesive  force  of  the  substance, 
as  given  in  the  table,  by  the  cohesive  force  per  square  inch,  area  of 
cross  section  (60000  lbs.)  of  wrought  iron,  which  gives  the  cohesive 
force  of  1  square  inch  area  of  cross  section  of  the  substance  whose 
power  is  sought  to  be  ascertained,  and  the  product  of  1  square  inch 
thus  found,  multiplied  by  area  of  cross  section,  in  inches,  of  the  rod, 
piece,  or  bar  itself,  gives  the  cohesive  force  thereof. 

Alloys  having  a  tenacity  greater  than  the  sum  of  their  constituents, 
Swedish  copper  6  pts.,  Malacca  tin  1 ;  tenacity  per  sq.  inch,  64000  lbs. 
Chili  copper  6  pts.,  Malacca  tin  1 ;  "  "         "     60000  " 

Japan  copper  5  pts.,  Banca  tin  1  ;  "  "         M     57000  " 

Anglesea  copper  6  pts.,  Cornish  tin  I ;  ••  "        M     41000  M 


LINEAR    DILATION    OF    SOLIDS    BY   HEAT. 


75 


Common  block-tin  4  pts.,  lead  1,  zinc  1 ;  tenacity  per  sq.  in.,  13000  lbs. 
Malacca  tin  4  pt8.,tegulu8  of  antimony  1;  "  "      "     12000" 

Block-tin  3  pts.,  lead  l  part;  "  "      «'     10000  " 

Block-tin  8    pts. ,  zinc  1  part;  "  "       "     10200" 

Zinc  1  part,  lead  1  purl;  «  "        M       4500  " 

A  Hoys  having  a  density  greater  than  the  mean  of  their  constituents. 

Cold  with  antimony,  bismuth,  cobalt,  tin,  or  zinc. 

Silver  with  antimony,  bismuth,  lead,  tin,  or  zinc. 

Copper  with  bismuth,  palladium,  tin,  or  zinc. 

Lead  with  antimony. 

Platinum  with  molybdinum. 

Palladium  with  bismuth. 

Alloys  having  a  density  less  than  the  mean  of  their  constituents. 
Gold  with  copper,  iron,  iridium,  lead,  nickel,  or  silver. 
Silver  with  copper  or  lead. 
Iron  with  antimony,  bismuth,  or  lead. 
Tin  with  antimony,  load,  or  palladium. 
Nickel  with  arsenic. 
Zinc  with  antimony. 

RELATIVE    POWER    OF    DIFFERENT    METALS    TO    CONDUCT   ELEC-. 

TRICITY, 

{the  mass  of  each  being  equal.) 

Copper,  ....  1000 1  Platinum,         .         .        .188 

Gold,  ....  936 1  Iron,         .         .         .         .158 

Silver,  ....  736 1  Tin,         .         .         .         .155 

Zinc,  ....  285  (Lead,       ....        83 


LINEAR    DILATION   OF    SOLIDS    BY    HEAT. 

Length  which  a  bar  heated  to  212°  has  greater  than  when  at  the  tem- 
perature of  32°. 


Brass,  cast, 

.     .0018671 

Iron,  wrought,    . 

.0012575 

Copper, 

.     .0017674 

Lead, 

.0028568 

Fire  brick, 

.     .0004928 

Marble,      . 

.0011016 

Glass, 

.     .0008545 

Platinum,  . 

.0009342 

Gold, 

.     .0014880 

Silver, 

.0020205 

Granite,     . 

.     .0007894 

Steel, 

.0011898 

Iron,  cast, 

.     .0011111 

Zinc, 

.0029420 

Note.  — To  find  the  surface  dilation  of  any  particular  article,  double  its  linear  dilation, 
and  to  find  the  di  ation  in  volume,  triple  it.  To  find  the  elongation  in  linear  inches  per 
linear  foot,  of  an)  particular  article,  multiply  its  respective  linear  dilation,  as  given  in  tlie 
by  12. 


76 


EFFECTS   OF  HEAT. 


MELTING    POINT    OF    METALS    AND    OTHER    BODIES. 
Lime,  palladium,  platinum,  porcelain,  rhodium,  silex,  may  be  melted 
by  means  of  strong  lenses,  or  by  the  hydro-oxygen  blowpipe.     Co- 
balt,  manganese,  plaster  of  Paris,  pottery \  iron,  nickel,  &c,  at  from 
2700°   to  3250°   Fahrenheit;  others  as  follows :  — 


Degree*  Pah. 

Degrees  Fab. 

Antimony, 

.     809 

Nitre, 

.     660 

Beeswax,  bleached,    . 

.     155 

Silver, 

.  1873 

Bismuth,    . 

.     506 

Solder,  common, 

.     475 

Brass, 

,              . 

.  1900 

"      plumbers', 

.     360 

Copper, 

„ 

.  1996 

Sugar, 

.     400 

Glass,  flint, 

„ 

.  1178 

Sulphur,    . 

.     226 

Gold, 

t             . 

.  2016 

Tallow,      . 

.     127 

Lead, 

1 

.     612 

Tin,. 

.     442 

Mercury,    . 

. 

.  —39 

Zinc, 

.     680 

Cast  iron  thoroughly  melts  at 

2786 

Greatest  heat  of  a  smith's  i 

brge,  (com.)   . 

2346 

Welding  heat  of  iron, 

.         .         . 

1892 

Iron  red  hot  in  twilight, 

884 

Lead  1,  tin  I,  bismuth  4, melts  at 

.    201 

Lead  2,  ti»  3,  bismuth  5, 

<i     it. 

.   212 

RELATIVE    POWER    OF    DIFFERENT     BODIES   TO    RADI/ 

kTE    HEAT. 

Water,     . 

100 

Lead,  bright,    . 

19 

Copper,    .... 

12 

Mercury, 

20 

Glass,      .... 

90 

Paper,  white,  . 
Silver,     . 

100 

Ice, 

85 

12 

India  ink, 

88 

Tin,  blackened, 

.       100 

Iron,  polished, 

15 

"      clean, 

12 

Lampblack, 

100 

"      scraped,  . 

16 

Nora.  —  The  power  of  a  body  to  rejlect  heat  is  inverse  to  its  power  of  radiation. 

BOILING   POINT    OF    LIQUIDS. 
Barometer  at  30  in. 


Acid,  nitric, 

253° 

Oils,  essential,  avg 

.,       .         318° 

"      sulphuric,     . 

600° 

"      turpentine, 

316? 

Alcohol,  anhyd.,    . 

168.5c 

11      linseed, 

640° 

"        90  per  cent.,    . 

174o 

Phosphorus, 

554^ 

Ether,  sulph., 

97° 

Sulphur, 

560° 

Mercury, 

656° 

Water, 

212° 

Nora  —  Barometer  at  31  inches,  water  boils  al213°57;  at  20,  it  boil*  at210°.3S;  at 
88,  it  boils  at  2080.69 ;  at  27,  it  boils  at  206°. 85,  and  in  vacuo  ii  tx>ils  at  88°.  No  liquid, 
■nder  pressure  of  the  atmosphere  alone,  can  be  heated  above  its  boiling  point.  At  that 
point  the  steam  emitted  sustains  the  weight  of  the  atmosphere. 


EFFECTS   OF   HEAT,    ETC. 


77 


FREEZING    POINT    OF    LIQUIDS. 


Acid,  nitric, 

.       —55° 

Oil,  linseed,  avg., 

—11° 

"      sulphuric, 

1° 

Proof  spirits, 

—7° 

Ether,  . 

.       —47° 

Spirits  turpentine, 

16° 

Mercury, 

.       —39° 

Vinegar, 

28° 

Milk,    . 

30° 

Water, 

32° 

Oil,  cinnamon, 

30° 

Wine,  strong,         . 

20° 

"    fennel,    . 

14° 

Rapeseed  Oil, 

25* 

**    olive, 

36° 

Note.  —  Water  expands  in  freezing  .11,  or  i  its  bulk. 

EXPANSION  OF  FLUIDS  BY  BEING    HEATED    FROM    32°    TO   212?,  P. 

Atmospheric  air,  3-^  per  each  degree,  =  .375 

Gases,  all  kinds,  Tfo  "       "         " 

Mercury,  exposed,     ......         .018 

Muriatic  acid,  (sp.  gr.  1.137,)  060 

Nitric  acid,  (sp.  gr.  1.40,) 110 

Sulphuric  acid,  (sp.  gr.  1.85,) 060 

"  ether,  —  to  its  boiling  point,        .         .         .070 

Alcohol,  (90  per  cent.,)         "         "  .         .         .110 

Oils,  fixed, 080 

"  turpentine, 070 

Water,      .     • 046 


RELATIVE  POWER  OF  SUBSTANCES  TO  CONDUCT  HEAT. 

363 

304 

180 

12 

11 

Note.  —  Different  woods  have  a  conducting  power  in  ratio  to  each  other,  as  is  Iheic 
respective  specific  gravities,  the  more  dense  having  the  greater. 


Gold, 

.      1000 

Zinc, 

Silver,    . 

973 

Tin, 

Copper,  . 

898 

Lead, 

Platinum, 

381 

Porcelain, 

Iron, 

374 

Fire  brick, 

METALS    IN    ORDER    OF    DUCTILITY   AND   MALLEABILITY. 


Ductility. 

1.  Platinum. 

2.  Gold. 

3.  Silver. 

4.  Iron. 

5.  Copper. 

6.  Zinc. 

7.  Tin. 

8.  Lead. 


Malleability. 

1.  Gold. 

2.  Silver. 

3.  Copper. 

4.  Tin. 

5.  Platinum. 

6.  Lead. 

7.  Zinc. 

8.  Iron. 


7* 


78 


MUTKITI'VK   AND  ALCOHOLIC   PROPERTIES   OF   BODIES. 


Quantity  per  cent,  by  weight  of  Nutritious  Matter  contained  in  different 
articles  of  Food. 


Articles. 

per  ct. 

Article*. 

perct. 

Lentils,    .... 

Oats, 

74 

Peas, 

93 

Meats,  avg.,     . 

35 

Beans, 

92 

Potatoes,  . 

25 

Corn,  (maize,) 

89 

Beets, 

14 

Wheat,    . 

85 

Carrots,    . 

10 

Barley,     . 

83 

Cabbage, . 

7 

Rice, 

88 

Greens,    . 

6 

Rye, 

i 

79 

Turnips,  white, 

4 

Specific  gravity,  and  quantity  per  cent.,  by  volume,  of  Absolute  Alcohol 
contained,  necessary  to  constitute  the  following  named  unadulterated 
articles. 

-  — -J-  Sp.  e-rar.        Per   cent. 

Art,cles-  60*.  6.  30.       ofAloohoF. 

Absolute  Alcohol,  (anhydrous,)  .  .  .  .7939  100 

Alcohol,  highest  by  distillation,  .  .  .  .825  92.6 

•■        commercial  standard,  .  .  .  .8335         90 

Proof  Spirits,  —  standard,  .  .  .  .9254         54 

Quantity  per  cent.,  by  volume,  (general  average)  of  Absolute  Alcohol 
contained  in  different  pure  or  unadulterated  Liquors,  Wines,  $c. 

per  cent. 

22 
20 
18 
17 
10 
16 
14 
12 
17 
19 

Proof  of  Spirituous  Liquors. 

The  weight,  in  air,  of  a  cubic  inch  of  Proof  Spirits,  at  60°  F.,  is 
233  grains ;  therefore,  an  inch  cube  of  any  heavy  body,  at  that  tempera- 
ture, weighing  233  grains  less  in  spirits  than  in  air,  shows  the  spirits 
in  which  it  is  weighed  to  be  proof.  If  the  body  lose  less  of  its  weight, 
the  spirit  is  above  proof,  —  if  more,  it  is  below. 


Liquor*,  4c. 

peT  cent. 

Wine*. 

Rum, 

50 

Port, 

Brandy,  . 

50 

Madeira, 

Gin,  Holland,  . 

48 

Sherry, 

Whiskey,  Scotch, 

50 

Lisbon,    . 

"        Irish, 

50 

Claret,     . 

Cider,  whole, 

9 

Malaga,    . 

Ale, 

8 

Champagne, 

Porter, 

7 

Burgundy, 

Brown  Stout,   . 

6 

Muscat, 

Perry,      . 

9 

Currant, 

COMPARATIVE    WEIGHT    OF    TIMBER. 


79 


Comparative  Weight  of  different  kinds  of  Timber  in  a  green  and  per' 
feclly  seasoned  state. 
Assuming  the  weight  of  each  kind  destitute  of  water  to  be  100,  that 
of  the  same  kind  green  is  as  follows :  — 


Ash, 

Beech, 

Birch, 


153 
174 

169 


Cedar,   . 
Elm,  swamp, 
Fir,  Amer., 


148  I  Maple,  red,  .  149 
198  I  Oak,  Am.,  .  151 
171  I  Pine,  white,  .       152 


Notb.  —  Woods  which  have  been  felled,  cleft  and  housed  for  12  months,  still  retain 
from  90  to  25  per  cent,  of  water.  They  therefore  contain  but  from  75  to  80  per  cent,  of 
healing  matter ;  and  it  will  require  from  23  to  29  per  cent,  the  weight  of  such  woods  to 
dispel  the  water  they  contain.  They  are,  therefore,  less  valuable  hy  weight,  as  fuel,  by 
this  per  cent.,  than  woods  perfectly  free  from  moisture.  They  never,  however,  contain, 
exposed  to  an  ordinary  atmosphere,  less  than  10  per  cent,  of  water,  however  long  kept; 
and  even  though  rendered  anhydrous  by  a  strong  heat,  they  again  imbibe,  on  exposure  to 
the  atmosphere,  from  10  to  12  per  cent,  of  dampness. 

Relative  power  of  different  seasoned  Woods,  Coals,  ^-c,  as  fuel,  to  pro- 
duce heat, — the  Woods  supposed  to  be  seasoned  to  mean  dryness, 
(77£  per  cent.,)  and  the  other  articles  to  contain  but  their  usual  quan- 
tity of  moisture. 


Hickory,  shell-bark, 
1 '         red-heart, 

Walnut,  com. 

Beech,  red, 

Chestnut, 

Elm,  white, 

Maple,  hard, 

Oak,  white, 
"      red,      . 

Pine,  white, 
"  yellow, 

Birch,  black, 
"      white, 

Coal,  Cumberland,  (bit.) 
"     Lackawanna  (anth.) 
"     Lehigh,  " 

"     Newcastle,  (bit.) 
11     Pictou,  (bit.) 
"     Pittsburgh,  (bit.) 
"     Peach  Mountain,  (anth.) 

Charcoal, 

Coke,  Virginia,  natural, 
"     Cumberland, 

Peat,  ordinary,     . 

Alcohol,  common, 

Beeswax,  yellow, 

Tallow,      . 


Ratio  of  Heatin* 
Power  per  equal 

Bulk. 

Weight. 

1.00 

1.00 

.81 

.99 

.95 

.98 

.74 

.99 

.49 

.98 

.58 

.98 

.66 

.98 

.81 

.99 

.69 

.99 

.42 

1.01 

.48 

1.03 

.63 

.99 

.48 

.99 

2.56 

2.28 

2.28 

2.22 

2.39 

2.03 

2.10 

1.96 

2.21 

1.91 

1.78 

1.82 

2.69 

2.29 

1.14 

2.53 

1.89 

2.12 

1.31 

2.25 

.62 

2.02 

2.90 

3.10 

80 


ILLUMINATION. 


Notb.  —  By  help  of  the  preceding  table,  the  price  of  either  one  article  being  known, 
the  relative  or  par  value  of  either  other,  as  fuel,  may  be  readily  ascertained  :  —  Example  : 

Maple  (66)  :  $5.00  :  :  Pine  (42)  :  $3.18. 


ILLUMINATION— ARTIFICIAL. 
The  following  Table  shows  :  — 

1.  The  materials  and  methods  of  using  —  column  Materials. 

2.  The  comparative  maximum  intensity  of  light  afforded  by  each 
material,  used  or  consumed  as  indicated, — column  Intensities. 

3.  The  weight,  in  grains,  of  material  consumed  per  hour,  by  each 
method  respectively,  in  producing  its  respective  light,  or  light  of  in- 
tensity ascribed  —  column  Weight. 

4.  The  ratio  of  weight  required  of  each  material,  under  each  spe- 
cial method  of  consumption,  for  the  production  of  equal  lights  in 
equal  times  —  column  Ratios. 

Materials.  Intens.    Weisrht.    Ratios. 

Camphene  *  Paragon  Lamp,           ...  16.  853  1. 

Sperm  Oil.   Parker's  heating  Lamp,      .         .  11.  696  1.19 

"      Mech.  orCarcel     "           .         .  10.  815  1.53 

"         "      French  annular        "                                  5.  543  2.04 

"      Common  hand          "                                   1.  112  2.10 

Whale    "     pTd.,  P's  heating"           .         .             9.  780  1.63 

Woo:  Candles,  3's  or  4's,  15  in.  or  12  in.,      .             1.  125  2.35 

"         "         6's,  9  in.,       ....               .92  122  2.50 

4's,  13£  in.',    ....             1.  142  2.66 

4's,  13£  in.,    ....             1.  168  3.15 

dipped,  10's,  ....               .70  150  4.02 

mould,  10's,   ....               .66  132  3.75 

8's,  ....               .57  132  4.35 

"        "             "         6's,             ...               .79  163  3.87 

4's,  131  in.,        .         .             1.  186  3.49 

"  Coal  Gas,"  intensity  being         ...  1.  740 

Note.  —  The  consumption  of  1.43  cubic  feet  of  gas  per  hour,  gives  a  light  equal  to  one 
wax  candle, —  the  consumption  of  1.%  cubic  feet  per  hour,  a  light  equal  to  four  wax  can- 
dles, and  the  consumption  of  3  cubic  feet  per  hour,  a  light  equal  to  tea  wax  candles.  A 
cubic  foot  of  gas  weL'hs  518  grains. 

The  average  yield  of  hi  carbureted  hydrogen  —  Olefiant  gas  —  Coal  Gas,  obtained  from 
the  following  articles,  is  as  annexed. 

1  lb.  Bituminous  Coal, 4£  cubic  feet. 

1  lb.  Oil,  or  Oleine, 15     "        " 

1  II).  Tar, 12      " 

1  lb.  Rosin,  or  Pitch, 10     " 

A  pipe  whose  interior  diameter  is  i  inch,  will  supply  gas  equal  in  illuminating  power  to 
20  wax  candles. 


Sperm  " 
Stearine" 
Tallow," 


1  lb.  Camphene, 
1  lb.  Sperm  Oil, 
1  lb.  Whale    "  p'Pd. 


lyVpinU 
1*- 


ILLUMINATION.  81 

By  the  foregoing  table,  it  is  readily  seen  in  what  ratio  the  several 
intensities,  furnished  by  the  different  methods,  stand  one  to  another, 
—  that  the  French  annular  lamp,  for  instance,  has  a  maximum  power 
=  half  that  of  the  mechanical,  or  T^  that  of  the  camphene,  or  5 
wax  candles,  3  to  the  lb.,  —  that  the  camphene,  at  its  maximum 
power,  yields  an  intensity  equal  to  that  afforded  by  16  -j-  .57,  »  28 
tallow  candles,  moulds,  8  to  the  lb.,  —  that  as  the  intensity  of  a  six 
wax  candle,  13  in.,  is  .92,  and  that  of  an  eight  mould  tallow  .57,  57 
candles  of  the  former  yield  an  intensity  equal  to  that  afforded  by  92 
of  the  latter,  &c,  &c. 

The  quantity  of  material  consumed  in  any  given  time  by  either 
of  the  foregoing  methods,  in  the  production  of  any  given  intensity  of 
light,  is  readily  ascertained  by  help  of  the  preceding  table.  Suppose, 
for  example,  an  intensity  equal  to  that  afforded  by  1  camphene  para- 
gon lamp  at  its  greatest  power,  is  required,  and  for  three  hours,  and 
that  it  is  proposed  to  produce  the  same  by  tallow  candles,  moulds,  10 
to  the  lb. ;  the  quantity  by  weight  of  candles  consumed  in  the  produc- 
tion is  required,  and,  consequently,  the  number  of  lights  that  must 
be  used. 

ILLUSTRATION. 
Intens.  of  camph.,  (16)  ■—  intensity  of  candles,  (.66)  =  24  candles,  and  grs.  in  1  h.  by 
1  candle,  (132)  X  24  X  3  hours  =  1  lb.  5f  oz.    Ana. 

The  economy  of  use,  as  between  any  two  materials,  under  either 
their  respective  forms,  or  methods  of  consumption,  for  the  production 
of  equal  lights  in  equal  times,  and  therefore  for  the  production  of  any 
intensity,  is  also,  by  help  of  the  given  table,  easily  learned,  the 
market  price  of  both  being  known  ;  and,  thereby,  the  per  cent.,  if  any 
difference  exist,  in  favor  of  the  more  economical,  or  less  expensive 
of  the  two,  may  be  found.     To  illustrate  :  — 

1.  The  price  of  camphene  is  10  cents  a  pound,  and  that  of  sperm 
oil,  15  ;  the  economy  of  use  as  between  the  two  for  the  production  of 
equal  lights  —  equal  intensities  in  equal  times,  greater  or  less  —  the 
former  consumed  in  the  paragon  lamp,  and  the  latter  in  Parker's  heat- 
ing, is  desired,  and  the  per  cent,  in  favor  of  the  less  expensive. 

10X1=  10,  and  15  X  1.19=  17.85;  showing  the  economy  to 
be  in  favor  of  the  camphene  —  showing  it  so  to  an  extent  17.85  —  10 
=  tW^*  or  to  an  extent  7  cts.  8j  mills  per  17  cts.  8  J  mills  —  to 
an  extent,  therefore, 

17.85  :  7.85  :  :  100  =  44  per  cent.     Ans. 

2.  The  price  of  sperm  candles,  4's,  is  40  cts.  a  pound,  and  the  price 
of  tallow,  mould  10's,  is  11  cts.  It  is  desired  to  know  which  of  the 
two,  for  the  production  of  an  intensity  nearest  obtainable  to  that  af- 
forded by  one  of  the  wax,  but  not  less  than  that  of  1  wax,  is  the  less 
expensive,  and  to  what  extent  per  cent. 

By  casting  the  eye  to  the  table,  it  is  readily  seen  that  two  tallow 
candles    must  be    employed,  the  comparative    intensity  of  which 


82  THERMOMETERS. 

is  .66  each  ;  therefore,  .66  X  2  =  1.32,  and  3.75  X  1.32  =  4.95, 
equivalent  weight ;  consequently  — 

40  X  2.66  =  106.4,  and  11  X  4.95  =  54.45,  and  106.4  —  54.45  == 
51.95.     Therefore, 

106.4  :  51.95  : :  lOO  =  48-^  per  cent.     Arts. 

Showing  that  an  intensity  nearly  J  greater  is  afforded  by  the  tallow 
than  the  wax,  and  at  an  expense  49  per  cent.  less.  The  same  rule 
of  practice  is  applicable,  as  between  any  two  methods,  for  equal  or 
greater  or  less  intensities,  as  desired. 


THERMOMETERS. 


Boiling  point. 

212° 

80° 

100° 


Freezing  point. 

32s 
0° 
0° 


Fahrenheit's,  . 

Reaumur's,      .... 

Centigrade,     .... 

To  reduce  Reaumur  to  Fahrenheit. 
When  it  is  desired  to  reduce  the  -\-°,  (degrees  above  the  zero)  :  — 

Rule.  —  Multiply  the  degrees  Reaumur,  by  2.25,  and  add  32°  to 
the  product ;  the  sum  will  be  the  degrees  Fahrenheit. 

When  it  is  desired  to  reduce  the  —  °,  (degrees  below  the  zero)  :  — 

Rule.  —  Multiply   the  —  °  Reaumur  by   2.25,  and   subtract  the 
product  from  32°  ;  the  difference  will  be  the  degrees  Fahrenheit. 

Example.  —  The  degrees  R.   are  40  ;   required  the  equivalent 
degrees  F. 

40  X  2.25  =  90  +  32  =  122°.     Ans. 

Example.  —  The  degrees  below  0,  R.,  are  10  ;  what  are  the  cor- 
responding degrees  F.  1 

10  X  2.25  =  22.5,  and  32  —  22.5  —  9£°.     Ans. 

'  Example.  —  Tbfl  degrees  below  0,  R.,  are  16  ;  what  point  on  the 
scale  F.  corresponds  thereto? 

16  X  2.25  =  36,  and  32  —  36  =  —  4  ;  4°  below  0.     Ans. 

1\>  reduce  the  Cmtigrade  to  Fahrenheit. 
Rule.  —  Multiply  the  degrees  C.  by  1.8,  and  in  all  other  respects 
proceed  as  directed  for  Reaumur,  above. 

Note.  — The  zero  of  Wodgowood't  pyrometer  is  And  tt  the  tamNMtamof  banitd*hoC 

in  daylight,  =  1077O  f.f  ;m,i  ,.,„•],  degree  W.  equals  13U°  F.    The  instrument  is  not  con- 
sidered reliable,  and  is  but  litUe  used 


HORSE    POWER  —  ANIMAL   POWER — STEAM.  88 

HORSE  POWER. 

A  house-power,  in  machinery,  as  a  measure  of  force,  is  estimated 
equal  to  the  raising  of  33000  lbs.  over  a  single  pulley  one  foot  a 
minute,  =  550  lbs.  raised  one  foot  a  second,  =  1000  lbs.  raised  33  feet 
a  minute. 


ANIMAL  POWER. 

A  man  of  ordinary  strength  is  supposed  capable  of  exerting  a  force 
of  30  lbs.  for  10  hours  in  a  day,  at  a  velocity  of  2£  feet  a  second,  = 
75  lbs.  raised  1  foot  a  second. 

The  ordinary  working  power  of  a  horse  is  calculated  at  750  lbs.  for 
8  hours  in  a  day,  at  a  velocity  of  2  feet  a  second,  =  375  lbs.  raised 
1  foot  a  second,  =  5  times  the  effective  power  of  a  man  during  asso- 
ciated labor,  and  4  times  his  power  per  day  ;  and  as  machinery  may 
be  supposed  to  work  continually,  =  a  trifle  less  than  23  per  cent,  per 
day  of  a  machine  horse-power. 


STEAM. 

Table  exhibiting  the  expansive  force  and  various  conditions  of  steam 
under  different  degrees  of  temperature. 


Decrees  of 
heat. 

Pressure  in 
atmospheres. 

Density. 
Water  m  1. 

Volume. 
Water  as  1. 

Spec,  gravity. 
Air  as  1. 

Weight  of  a 

cubic   loot   in 

grains. 

212 

1 

.00059 

1694 

.484 

254 

250.5 

2 

.00110 

909 

.915 

483 

270 

3 

.00160 

625 

1.330 

700 

293.8 

4 

.00210 

476 

1.728 

910 

308 

5 

.00258 

387 

2.120 

1110 

359 

10 

.00492 

203 

3.970 

2100 

418.5 

20 

.00973 

106 

7.440 

3940 

[An  atmosphere  is  14^  lbs.  to  the  square  inch.] 

Note.  —  By  the  above  table  it  is  seen  that  any  given  quantity  of  steam  having  a  tem- 
perature of  212°  F.,  occupies  a  space,  under  the  ordinary  pressure  of  the  atmosphere, 
1694  times  greater  than  it  occupied  when  as  water  in  a  natural  state.  It  exerts  a  mechan- 
ical force,  consequently,  =  1694  times  the  weight  or  force  of  the  atmosphere  resting  on 
the  bulk  from  which  it  was  generated,  or  resting  on  l-1694th  of  the  space  it  occupies. 
A  force,  if  we  consider  the  volume  as  so  many  cubic  inches,  equal  to  the  raising  of  2087 
lbs.  12  inches  high,  by  a  quantity  of  steam  less  than  a  cubic  foot,  heated  only  to  the  tem- 
perature of  boiling  water,  and  weighing  but  248  grains,  and  that,  too,  the  product  of  a 
single  cubic  inch  of  water. 

The  mean. pressure  of  the  atmosphere  at  the  earth's  surface  is  equal 
to  the  weight  of  a  column  of  mercury  29.9  inches  in  height,  or  to  a 
column  of  water  33.87  feet  in  height,  =  2110.8  lbs.  per  square  foot,  or 


84 


VELOCITY   AND   FORCE   OF   WIND. 


14.7  lbs.  per  square  inch.  Its  density  above  the  earth  is  uniformly 
less  as  its  altitude  is  greater,  and  its  extent  is  not  above  50  miles  — 
its  mean  altitude  is  about  45  miles ;  at  44  miles  it  ceases  to  reflect 
light.  Were  it  of  uniform  density  throughout,  and  of  that  at  the  sur- 
face, its  altitude  would  be  but  54  miles.  Its  weight  is  to  pure  water 
of  equal  temperature  and  volume,  as  1  to  829.  It  revolves  with  the 
earth,  and  its  average  humidity,  at  40°  of  latitude,  is  4  grains  per  cubic 
foot.  Its  weight  at  60°,  b.  30,  compared  with  an  equal  bulk  of  pure 
water  at  40°,  b.  30,  is  as  1  to  830.1. 


VELOCITY  AND    FORCE  OF  WIND. 


Mean  ve 

locity  in 

1 

Miles  per 

Feet  per 

Force  in  lb».  per 

Appellations. 

hour. 

second. 

square  foot. 

Just  perceptible, 

2£ 

3| 

.032 

Gentle,  pleasant  wind, . 

4£ 

H 

.101 

Pleasant,  brisk  gale,     . 

m 

18} 

.80 

Very  brisk,         " 

22£ 

33 

2.52 

High  wind, 

32£ 

47| 

5.23 

Very  high  wind,  . 

42£ 

62J 

8.92 

Storm,  or  tempest, 

50 

73J 

12.30 

Great  storm, 

60 

88 

17.71 

Hurricane, 

80 

in* 

31.49 

Tornado,  moving  buildings,  &c, 

100 

146.7 

49.20 

The  curvature  of  the  earth  is  6.99  inches  (.5825  foot)  in  a  single 

statute  mile,  or  8.05  inches  in  a  geographical  mile,  and  is  as  the 

square  of  the  distance  for  any  distance  greater  or   less,  or   space 

between  two  levels  ;  thus,  for  three  statute  miles  it  is 

1  :  32  :  :  6.99  :  54  feet,  nearly. 

The  horizontal  refraction  is  T(r. 

Degrees  of  longitude  are  to  each  other  in  length,  as  the  cosines  of 
their  latitudes.  At  the  equator  a  degree  of  longitude  is  60  geographical 
miles  in  length,  at  90°  of  latitude  it  is  0 ;  consequently,  a  degree  of 


longitude  at 

5° 

=  59.77  miles. 

40° 

=  45.96  miles 

10° 

=  59.09     " 

50° 

=  38.57     " 

20° 

=  56.38     " 

70° 

=  20.52     " 

30° 

=  51.96     " 

85°        . 

=    5.23     " 

Time  is  to  longitude  4  minutes  to  a  degree,  —  faster,  east  of  any 
given  point ;  slower,  west. 

The  mean  velocity  of  sound  at  the  temperature  of  33°  is  1100  feet 
a  second.     Its  velocity  is  increased  £  a  foot  a  second  for  overy  degree 


GRAVITATION.  85 

above  33°,  and  decreased  £  a  foot  a  second  for  every  degree  below 
33  . 

In  water,  sound  passes  at  the  rate  of  4,708  feet  a  second. 

Light  travels  at  the  rate  of  192,000  miles  per  second. 

GRAVITATION. 

Gravity,  or  Gravitation,  is  a  property  of  all  bodies,  by  which 
they  mutually  attract  each  other  proportionally  to  their  masses, 
and  inversely  as  the  square  of  the  distance  of  their  centres  apart. 
Practically,  therefore,  with  reference  to  our  Earth  and  the  bodies 
upon  or  near  its  surface,  gravity  is  a  constant  force  centred  at  the 
Earth's  centre,  and  is  there  continually  operating  to  draw  all  bodies 
with  a  uniformly  accelerating  velocity  to  that  point,  and  through 
very  nearly  equal  spaces,  in  equal  intervals  of  time  from  rest,  at  all 
localities. 

Putting  R1  to  represent  the  Equatorial  radius  of  the  earth,  and  r 
to  represent  the  Polar,  and  making  IV  =  3962.5  statute  miles,  and 
r=  3949.5,  which  is  nearly  in  accordance  with  the  mean  of  the 
most  reliable  measurements  of  the  arcs  of  a  degree  of  latitude  at 
different  locailties,  we  have  ea  =  (R12 —  r2)  -7-  R>2  =  .006550751 ,  the 
square  of  the  ellipticity  of  the  earth,  and  R  =  2R'  -7-  (2  -f-  e2  sinH), 
the  radius  at  any  given  latitude  I. 

And  since  the  initial  velocity  due  to  gravity  at  the  level  of  the 
sea  at  the  Equator  is  G  =  32.0741  feet  per  second,  or,  in  other 
words,  since  a  body  falling  in  vacuo  at  the  equator,  at  the  level  of 
the  sea,  describes  a  space  of  16.03705  feet  in  the  first  second  of 
time  from  rest,  we  have  g  =  [Rt  ^G)-^-RJi,  the  initial  velocity  at 
the  level  of  the  sea  at  any  given  radius  R ;  or  g  =  (22441.2  —  R)\ 

a     i  a     11  /  22441.2  V        /  2h    \ 

And  finally g={——)   x  (l -g^)  at  any  given   ra- 

dius  R,  at  any  given  altitude,  Ji,  in  feet,  above  the  level  of  the  sea. 

Note.  —  When  I,  reckoned  from  the  equator,  is  higher  than  45°,  sin2  I  =s 
cosa  ('JO  —  I). 

The  momentum,  or  force,  with  which  a  falling  body  strikes,  is  the 
product  of  its  weight  and  velocity  (the  weight  multiplied  by  the 
square  root  of  the  product  of  the  .space  fallen  through  and  64.33, 
or  4  times  16T^)  ;  thus,  100  lbs.,  falling  50  feet,  will  strike  with  a 

50  X  64.333=^3216.66  =  56.71  X  100  =  5671  lbs. 
An  entire  revolution  of  the  earth,  from  west  to  east,  is  performed 
in  23  hours,  56  minutes,  and  4  seconds.     A  solar  year  =  365  days, 
5  hours,  48  minutes,  57  seconds. 

The  area  of  the  earth  is  nearly  1 9  7,000,000  square  miles.    Its  crust 

is  supposed  to  be  about  30  miles  in  thickness,  and  its  mean  density  5 

times  that  of  water.     About  f  of  its  area,  or  150,000,000  square 

miles,  is  covered  by  water.    The  portions  of  land  in  the  several 

8 


80  CHEMICAL  ELEMENTS. 

divisions,    in   square    miles,    are,   in    round    numbers,   as  follows, 
viz:  — 

Asia,            .  .  16,300,000 1  Europe,        .         .           3,700,000 

Africa,         .  .  11,000,000  Australia,    .         .           3,000,000 

America,     .  .  14,500,000 1 

America  is  9000  miles  long,  or  T^y  the  circumference  of  the 
earth. 

The  population  of  the  globe  is  about  1,000,000,000,  of  which  there 
are,  in 

Asia,  .         .         456,000,000  I  Africa,      .         .  62,000,000 

Europe,     .         .         258,000,000  |  America,  .         .  55,000,000 


CHEMICAL  ELEMENTS. 

The  chemical  elements  —  simple  substances  in  nature  —  as  far  as 
has  been  determined,  are  58  in  number :  13  non-metallic  and  45 
metallic. 

Of  the  non-metallic,  5 — bromine,  chlorine,  fluorine,  iodine,  and  oxy- 
gen, (formerly  termed  "supporters  of  combustion,")  have  an  intense 
affinity  for  all  the  others,  which  they  penetrate,  corrode,  and  appar- 
ently consume,  always  with  the  production,  to  some  extent,  of  light 
and  heat.  They  are  all  non-conductors  of  electricity  and  negative 
electrics. 

The  remaining  8  —  hydrogen,  nitrogen  or  azote,  carbon,  boron,  sili- 
con, phosphorus,  selenium,  and  sulphur,  are  eminently  susceptible  of 
the  impressions  of  the  preceding  five  ;  when  acted  upon  by  either  of 
them  to  a  certain  extent,  light  and  heat  are  manifestly  evolved,  and 
they  are  thereby  converted  into  incombustible  compounds. 

Of  the  metals,  7 —  potassium,  sodium,  calcium,  barytium,  lithium, 
strontium,  and  magnesia,  by  the  action  of  oxygen,  are  converted  into 
bodies  possessed  of  alkaline  properties. 

Seven  of  them  —  glucinum,  irhium,  terbium,  yttrium,  allumiuni,  zir- 
conium, and  thorium,  —  by  the  action  of  oxygen,  are  converted  into 
tin-  earths  proper. 

In  short,  all  the  metals  are  acted  upon  by  oxygen,  as  also  by  most 
or  all  of  the  non-metallic  family.  The  compounds  thus  formed  are 
alkaline,  saline,  or  acidulous,  or  an  alkali,  ■  salt,  or  BO  arid,  according 
to  the  nature  of  the  materials  and  the  extent  of  combination. 

Metals  combine  with  each  other,  forming  alloy*.    W  one  of  the 

metaU  in  combination  is  mercury,  the  compound  is  called  an  amalgam. 

Silicon  is  the  base  of  the  mineral  World,  and  carbon  of  the  organ- 
oid. 

For  a  very  general  list  of  the  metals,  see  Table  of  Specific  (Jrav- 
ities. 


CONSTITUENTS    OF    BODIES. 


87 


TABLE 

Exhibiting  the  Elementary  Constituents  and  per  cent,  by  weight  of  each, 
in  100  parts  of  different  compounds. 


Com  pound*. 

Constituents  nn>j  per  cent. 

Atmospheric  air,  ...       a 

Hydrogen. 

Oxysren. 

Azote. 

( 'in  DOB, 

20.8 

79.2 

AV;iter,  pure, 

11.1 

88.9 

Alcohol,  anhydrous, 

12.9 

34.44 

52.66 

Olive  oil,      .... 

13.4 

9.4 

77.2 

Sperm "        .         ... 

10.97 

10.13 

78.9 

Castor  " 

10.3 

15.7 

74.00 

Stearine,  (solid  of  fats,) 

11.23 

6.3 

0.30 

82.17 

Oleine,  (liquid  of  fats,) 

11.54 

12.07 

0.35 

76.03 

Linseed  oil,  . 

11.35 

12.64 

76.01 

Oil  of  turpentine, 

11.74 

3.66 

84.6 

"  Camphene,"  (pure  spts.  turp.) 

11.5 

88.5 

Caoutchouc,  (gum  elastic,)     . 

10. 

90. 

Camphor,      .... 

11.14 

11.48 

77.38 

Copal,  resin, 

9. 

11.1 

79.9 

Guaiac,  resin, 

7.05 

25.07 

67.88 

Wax,  yellow, 

11.37 

7.94 

80.69 

Coals,  cannel, 

3.93 

21.05 

2.80 

72.22 

"      Cumberland, 

3.02 

14.42 

2.56 

80. 

"      Anthracite,          .         .       b 

93. 

Charcoal,      .... 

97. 

Diamond,      .... 

100. 

Oak  wood,  dry,                                c 

5.69 

41.78 

52.53 

Beech  "       " 

5.82 

42.73 

51.45 

Acetic   acid,   dry, 

5.82 

46.64 

47.54 

Citric       "      crystals,  . 

4.5 

59.7' 

35.8 

Oxalic     "     -dry, 

79.67 

20.33 

Malic,      "      crystals,    . 

3.51 

55.02 

41.47 

Tartaric  "      dry, 

3. 

60.2 

36.80 

Formic    " 

2.68 

64.78 

32.54 

Tannin,  tannic  acid,  solid, 

4.20 

44.24 

51.56 

Nitric  acid,  dry,    .         . 

73.85 

26.15 

Nitrous  "     anhydrous,  liquid, 

61  32 

30.68 

Ammoniacal    gas, 

17.47 

82.53 

Carbonic  acid    "    . 

72.32 

27.68 

Carb.  hydrogen  gas, 

24.51 

75.49 

Bi-carb.  hyd.,  okfient  gas,     . 

14.05 

85.95 

Cyanogen                       " 

53.8 

46.2 

Nitric  oxyde                  " 

53. 

47.00 

Nitrous  "                      " 

36.36 

63.64 

Ether,  sulphuric,  . 

13.85 

21.24 

65.05 

Creosote,      .... 

7.8 

16. 

76.2 

88 


CONSTITUENTS   OF   BODIES. 


Compound!. 

L 
Hydro  g-en. 

onstituenti 
Oxygen. 

and  per  cen 
Azote. 

Carbon. 

Morphia,       .... 

6.37 

16.29 

5. 

72.34 

Quina,  —  quinine, 

7.52 

8.61 

8.11 

75.76 

Veratrine,     .... 

8.55 

19.61 

5.05 

66.79 

Indigo, 

4.38 

14.25 

10. 

71.37 

Silk,  pure  white, 

3.94 

34.04 

11.33 

50.69 

Starch,  —  farina,  < 

lextrine, 

6.8 

49.7 

43.5 

Sugar, 

. 

6.29 

50.33 

43.38 

Gluten, 

,         . 

7.8 

22. 

14.5 

55.7 

Wheat, 

, 

c 

6. 

44.4 

2.4 

47.02 

Rye,     . 

. 

5.7 

45.3 

1.7 

47.03 

Oats,    . 

,         # 

6.6 

38.2 

2.3 

52.9 

Potatoes, 

# 

6.1 

46.4 

1.06 

45.9 

Peas,    . 

g 

6.4 

41.3 

4.3 

48. 

Beet  root,     . 

§ 

6.2 

46.3 

1.8 

45.7 

Turnips, 

. 

6. 

45.9 

1.8 

46.3 

Fibrin, 

. 

'.       d 

7.03 

20.30 

19.31 

53.36 

Gelatin, 

, 

.       d 

7.91 

27.21 

17. 

47.88 

Albumen, 

• 

.       d 

7.54 

23.88 

15.70 

52.88 

Muriatic  acid  gas,  —  Hydrogen  5.53  -|-  94.47  chlorine. 
Sulphuric  acid,  dry,  — Oxygen  79.67  -\-  20.33  sulphur. 
Silicic  acid  —  Silica,  dry,  —  Oxygen  51.96  -J-  48.04  silicon. 
Boracic  acid  — Borax,  dry,—      "        68.81  -f-  31.19  boron. 

a.  The  atmosphere,  in  addition  to  its  constituents  as  given  in  the 
table,  contains,  besides  a  small  quantity  of  vapor,  from  1  to  3  parts  in 
a  thousand  of  carbonic  acid  gas,  and  a  trace  merely  of  ammoniacal  gas. 

b.  Anthracite  coal,  charcoal,  plumbago,  coke,  &c,  have  no  other 
constituent  than  carbon  ;  they  are  combined,  to  a  small  extent,  with 
foreign  matters,  such  as  iron,  silica,  sulphur,  alumina, '&c. 

c.  The  constituents  of  woods,  grains,  &c,  are  given  per  cent.,  with- 
out regard  to  the  foreign  matters  (metallic)  which  they  contain.  In 
oak,  chestnut,  and  Norway  pine,  the  ashes  amount  to  about  -fa  of  1  per 
cent.,  and  in  ash  and  maple  to  -^  of  1.  In  anthracite  coals,  at  an 
average,  they  are  about  7  per  cent. 

d.  Fibrin,  Gelatin,  Albumen  —  Proximate  animal  constituents  — 
Nutritious  properties  of  animal  matter. 

Fibrin  is  the  basis  of  the  muscle  (lean  meat)  of  all  animals,  and  is 
also  a  large  constituent  of  the  blood. 

Gelatin  exists  largely  in  the  skin,  cartilages,  ligaments,  tendons  and 
bones  of  animals.      It  also  exists  in  the  muscles  and  the  membram  s. 

Albumen  exists  in  the  skin,  glands  and  vessels,  and  in  the  serum  of 
the  blood.     It  constitutes  nearly  the  whole  of  the  white  of  an  egg. 


CONSTITUENTS   OP   BODIES.  89 

The  relative  quantities  by  volume  of  the  several  gases  going  to 
constitute  any  particular  compound,  are  readily  ascertained  by  help 
of  their  respective  specific  gravities,  compared  with  their  relative 
weights,  as  given  per  cent,  in  the  preceding  table:— thus,  the  sp. 
gr.  of  hydrogen  is  .0689,  and  that  of  oxygen  1.1025,  and  1.1025  -J- 
.0689  =  16 ;  showing  the  weight  of  the  latter  to  be  16  times  that  of 
the  former  per  equal  volumes,  or,  relatively,  as  16  to  1.  The  per 
cent,  by  weight,  as  shown  by  the  table,  in  which  these  two  gases 
combine  to  form  water,  for  instance,  is  11.1  and  88.9  ;  or  11.1  of 
hydrogen  and  88.9  of  oxygen  in  100  of  the  compound  ;  or  as  88.9  -j- 
11.1,  —  as  8  to  1:  16 -j- 8  =  2  :  two  volumes,  therefore,  of  the 
lighter  gas  (hydrogen)  combine  with  one  of  oxygen  to  form  water. 
Water,  consequently,  ts  a  Protoxide  of  Hydrogen. 

Upon  the  principle  of  atomic  weights  —  primal  quantities,  by 
weight,  in  which  bodies  combine,  based  upon  some  fixed  radix,  usually 
hydrogen  as  it  forms  with  water,  and  as  1,  —  we  have,  for  water, — 
II1  -\-  O*  =  Aq.  9.  An  atom  of  hydrogen,  therefore,  is  1,  an  atom 
of  oxygen  8,  and  an  atom  of  water  9. 

By  the  same  rule  as  the  preceding,  the  constituents  of  atmospheric 
air  are  found  to  be  to  each  other  in  volume  as  4  to  1,  — 4  volumes  of 
nitrogen  and  1  volume  of  oxygen  =»  atmospheric  air.  The  weight 
of  nitrogen  to  hydrogen  per  equal  volumes,  is  as  14.14  to  1.  Atomic- 
ally,  therefore,  oxygen  being  8,  it  is  as  7.07  to  1 ;  hence  we  have 
N4  -j-  O  =  36.28,  the  atomic  weight  of  atmosphere. 

The  vast  condensation  of  the  gases  which  takes  place,  in  some  in- 
stances, in  forming  compounds,  may  be  conceived  of,  and  the  process 
for  determining  the  same  exhibited  by  a  single  illustration.  We  will 
take,  for  example,  water.  A  single  cubic  inch  of  distilled  water,  at 
60°,  weighs  252.48  grains.  Its  weight  is  to  that  of  dry  atmosphere, 
at  the  same  temperature,  as  827.8  to  1.  A  cubic  inch  of  dry  atmos- 
phere, therefore,  at  that  density,  weighs  .305  of  a  grain.  Hydrogen, 
we  find  by  the  table  of  Specific  Gravities,  weighs  .0689  as  much 
as  atmosphere,  and  oxygen  1.1025  as  much.  A  cubic  inch  of  hydro- 
gen, therefore,  weighs  .0689  X  .305  =  .0210145  of  a  grain,  and 
a  cubic  inch  of  oxygen  1.1025  X  .305,=  .3362625  of  a  grain. 
The  constituents  of  water  by  volume  are  2  of  the  first  mentioned  gas 
to  1  of  the  hitter;  and  .0210145  X  2  -f  .3362625  =  .3782915  of  a 
grain,  =  weight  of  three  cubic  inches  of  the  uncondensed  compound, 
]  of  which,  .1260972  of  a  grain,  is  the  weight  of  a  volume  1  cubic 
inch. 

As  the  weight  of  a  given  volume  of  the  uncondensed  compound,  is 
to  the  weight  of  an  equal  volume  of  the  condensed  compound,  so  are 
their  respective  volumes,  inversely  :  then  — 

.1260972  :  252.18  ::  1  :  2002.26,  the  number  of  cubic  inches  of  the 
two  gases  condensed  into  1  inch  to  form  water ;  a  condensation  of 
2001  times.  Of  this  volume  of  gases,  §,  or  1334.84  cubic  inches,  is 
hydrogen  ;  the  remaining  third,  667.42  cubic  inches,  is  oxygen. 

8* 


90  PROPERTIES,    ETC.,  OF   BODIES. 

The  foregoing  method,  though  strictly  correct,  does  not  exhibit  in  a 
general  way  the  most  expeditious  for  solving  questions  of  that  nature, 
the  condensation  which  takes  place  in  the  gases  on  being  converted 
into  solids,  or*  dense  compounds.  It  was  resorted  to,  in  part,  as  a 
means  through  which  to  exhibit  principles  and  proportions  pertaining 
thereto. 

As  before ;  one  cubic  inch  of  water  weighs  252.48  grains,  .*.  of 
which,  or  28.05-}-  grains,  is  hydrogen,  and  |^  or  224. 43 —  grains,  is 
oxygen.  The  volume  of  1  grain  of  oxygen  is  2.97-j-  cubic  inches,  and 
the  volume  of  hydrogen  is  16  times  as  much,  or47.58-|-  cubic  inches. 
Therefore,  28.05  X  47.58  =  1334.62,  and  224.43  X  2.97  =  665.56, 
*=  2001.18,  condensation,  as  before. 


Properties  of  tlie  simple  substances,  and  some  of  their  compounds,  not 
given  in  the  foregoing. 
Bromine,  —  at  common  temperatures,  a  deep  reddish-brown  vola- 
tile liquid  ;  taste  caustic  ;  odor  rank  ;  boils  at  116°  ;  congeals  at  4° ; 
exists  in  sea-water,  in  many  salt  and  mineral  springs,  and  in  most 
marine  plants  ;  action  upon  the  animal  system  very  energetic  and 
poisonous  —  a  single  drop  placed  upon  the  beak  of  a  bird  destroys  the 
bird  almost  instantly.  A  lighted  taper,  enveloped  in  its  fumes,  burns 
with  a  flame  green  at  the  base  and  red  at  the  top  ;  powdered  tin  or 
antimony  brought  in  contact  is  instantly  inflamed  ;  potash  is  exploded 
with  violence. 

Chlorine,  —  a  greenish-yellow,  dense  gas  ;  taste  astringent;  odor 
pungent  and  disagreeable  ;  by  a  pressure  of  60  lbs.  to  the  square  inch 
is  reduced  to  a  liquid,  and  thence,  by  a  reduction  of  the  temperature 
below  32°,  into  a  solid.  It  exists  largely  in  sea-water — is  a  constit- 
uent of  common  salt,  and  forms  compounds  with  many  minerals ;  is 
deleterious,  irritating  to  the  lungs,  and  corrosive ;  has  eminent 
bleaching  properties,  and  is  the  greatest  disinfecting  agent  known  ; 
a  lighted  taper  immersed  in  it  burns  with  a  red  flame ;  pulverized 
antimony  is  inflamed  on  coming  in  contact,  so  is  linen  saturated  with 
oil  of  turpentine  ;  phosphorus  is  ignited  by  it,  and  burns,  while  im- 
mersed, with  a  pale-green  flame  ;  with  hydrogen,  mixed  measure  for 
measure,  it  is  highly  explosive  and  dangerous. 

Fluorine,  —  a  gas,  similar  to  chlorine,  —  exists  abundantly  in 
fluor-spar. 

Oxygen, — a  transparent,  colorless,  tasteless,  inodorous,  innoxious 
gas  ;  supports  respiration  and  combustion,  but  will  not  sustain  life  for 
any  length  of  time,  if  breathed  in  a  pure  state.  It  is  by  far  the  most 
abundant  substance  in  existence ;    constitutes  ^  of  the  atmosphere  ; 


PROPERTIES,    ETC.,  OF   BODIES.  01 

g  of  water ;  and  nearly  the  whole  crust  of  the  earth  is  oxidized  sub- 
stances. For  further  combinations  and  properties,  see  tables  of  Ele- 
mentary Constituents  and  Chemical  Elements. 

Iodine, — at  common  temperatures,  a  soft,  pliable,  opaque,  bluish- 
black  solid  ;  taste  acrid  ;  odor  pungent  and  unpleasant ;  fuses  at  225°  ; 
boils  at  317°  ;  its  vapor  is  of  a  beautiful  violet  color  ;  it  inflames 
phosphorus,  and  is  an  energetic  poison  ;  exists  mainly  in  sea-weeds 
and  sponges. 

Hydrogen,  —  a  transparent,  colorless,  tasteless,  inodorous,  innox- 
ious gas  ;  if  pure,  will  not  support  respiration  ;  if  mixed  with  oxy- 
gen, produces  a  profound  sleep  ;  exists  largely  in  water  ;  is  the  basis 
of  most  liquids,  and  is  by  far  the  lightest  substance  known ;  burns  in 
the  atmosphere  with  a  pale,  bluish  light ;  mixed  with  common  air,  1 
measure  to  3,  it  is  explosive  ;  mixed  with  oxygen,  2  measures  to  1, 
it  is  violently  so. 

Nitrogen,  or  Azote,  —  a  transparent,  colorless,  tasteless,  inodorous 
gas ;  will  not  support  respiration  or  combustion,  if  pure  ;  exists 
largely  as  a  constituent  of  the  atmosphere  —  in  animals,  and  in  fun- 
gous plants  ;  is  evolved  from  some  hot  springs  ;  in  connection  with 
some  bodies,  appears  combustible. 

Carbon,  —  the  diamond  is  the  only  pure  carbon  in  existence  ;  pure 
carbon  cannot  be  formed  by  art ;  charcoal  is  97  per  cent,  carbon  ;  plum- 
bago, 95  ;  anthracite,  93.  Carbon  is  supposed  by  some  to  be  the  hard- 
est substance  in  nature.  A  piece  of  charcoal  will  scratch  glass;  but 
it  is  doubtful  if  this  is  not  due  to  the  form  of  its  crystals,  rather  than 
to  the  first  mentioned  quality.  It  is  doubtless  the  most  durable.  For 
combinations,  &c,  see  table. 

Boron,  — a  tasteless,  inodorous,  dark  olive-colored  solid. 

Silicon,  —  a  tasteless,  inodorous  solid,  of  a  dark-brown  color; 
exists  largely  in  soils,  quartz,  flint,  rock-crystal,  &c.  ;  burns  readily 
in  air  —  vividly  in  oxygen  gas  ;  explodes  with  soda,  potassa,  barryta. 

Phosphorus, — a  transparent,  nearly  colorless  solid,  of  a  wax- 
like texture  ;  fuses  at  108°,  and  at  550°  is  converted  into  a  vapor; 
exists  mainly  in  bones  —  most  abundant  in  those  of  man  —  is  poison- 
ous ;  at  common  temperatures  it  is  luminous  in  the  dark,  and  by  fric- 
tion is  instantly  ignited,  burning  with  an  intense,  hot,  white  flame ; 
must  be  kept  immersed  in  water. 

Selenium,  —  a  tasteless,  inodorous,  opaque,  brittle,  lead-colored 


92  PROPERTIES,   ETC.,  OF   BODIES. 

solid,  in  the  mass ;  in  powder,  a  deep-red  color  ;  becomes  fluid  at 
216°,  boils  at  050°;  vapor,  a  deep  yellow;  exists  but  sparingly, 
mainly  in  combination  with  volcanic  matter ;  is  found  in  small  quan- 
tities combined  with  the  ores  of  lead,  silver,  copper,  mercury. 

Ammoniacal  gas,  —  N  -f-  H3 ;  transparent,  colorless,  highly  pun- 
gent and  stimulating  ;  alkaline  ;  is  converted  into  a  transparent  liquid 
by  a  pressure  of  6.5  atmospheres,  at  50°  ;  does  not  support  respira- 
tion ;  is  inflammable. 

Carbonic  acid  gas,  —  C  -f-  O'2 ;  transparent,  colorless,  inodorous, 
dense  ;  is  converted  into  a  liquid  by  a  pressure  of  36  atmospheres  ; 
exists  extensively  in  nature,  in  mines,  deep  wells,  pits ;  is  evolved 
from  the  earth,  from  ordinary  combustion,  especially  from  the  combus- 
tion of  charcoal,  and  from  many  mineral  springs  ;  is  expired  by  man 
and  animals  ;  forms  44  per  cent,  of  the  carbonate  of  lime  called  mar- 
ble ;  the  brisk,  sparkling  appearance  of  soda-water,  and  most  mineral 
waters,  is  due  to  its  presence.  It  is  neither  a  combustible  nor  a  sup- 
porter of  combustion  ;  and,  when  mixed  with  the  atmosphere  to  an 
extent  in  which  a  candle  will  not  burn,  is  destructive  of  life.  Being 
heavier  than  atmosphere,  it  maybe  drawn  up  from  wells  in  large  open 
buckets ;  or  it  may  be  expelled  by  exploding  gunpowder  near  the  bot- 
tom.    Large  quantities  of  water  thrown  in  will  absorb  it. 

The  above  gas  is  expired  by  man  to  the  extent  of  1632  cubic  inches 
per  hour ;  it  is  generated  by  the  burning  of  a  wax  candle  to  the  ex- 
tent of  800  cubic  inches  per  hour  :  and,  by  the  burning  of  "Cam- 
phenc,"  (in  the  production  of  light  equal  to  that  afforded  by  1  \v;ix 
candle,)  to  the  extent  of  875  cubic  inches  per  hour.  Two  burning 
candles,  therefore,  vitiate  the  air  to  about  the  same  extent  as  1  per- 
son. 

Carbonic  oxide  gas,  —  C  -\-  O  ;  transparent,  colorless,  insipid  ; 
odor  offensive ;  does  not  support  combustion;  an  animal  confined  in 
it  soon  dies  ;  is  highly  inflammable,  burning  with  a  pale  blue  flame  ; 
mixed  with  oxygen,  1  to  2,  is  explosive  —  with  atmosphere,  even  in 
small  quantity,  is  productive  of  giddiness  and  fainting. 

Carburrtul  hydrogen  gas,  —  C  -j-  Ha ;  transparent,  colorless,  taste- 
less, nearly  inodorous;  exists  in  marshes  and  stagnant  pools  —  is 
there  formed  by  the  decomposition  of  vegetable  matter  ;  extinguishes 
all  burning  bodies,  but  at  the  same  time  is  itself  highly  eombustible, 
burning  with  a  bright  but  yellowish  flame  ;  it  is  destructive  to  life,  it 
respired. 

:  no  gen  —  Bkarburct  of  Nitrogen  —  a  gas,  — N  -f-  C  ;    trans 
:,  colorless,  highly  pungent  and  irritating  ;  under  a  pressure  of 


TROrERTIES,    ETC.,  OF   BODIES.  \)d 

3.6  atmosphen  s,  becomes  a  limpid  liquid  ;   burns  with  a  beautiful 
purple  flame. 

Hydrochloric  acid  gas —  Muriatic  acid  gas,  —  II  -j-  CI.  (chlorine)  ; 
transparent,  colorless,  pungent,  acrid,  suffocating  ;  strong  acid  taste. 

Nitrous  oxide  gas  —  Protoxide  of  Nitrogen,  "  laughing  gas," — 
N  -j-  O  ;  transparent,  colorless,  inodorous  ;  taste  sweetish  ;  powerful 
stimulant,  when  breathed,  exciting  both  to  mental  and  muscular  ac- 
tion ;  can  support  respiration  but  from  3  to  4  minutes  ;  is  often  per- 
nicious in  its  effects. 

Nitric  oxide  gas —  Binoxide  of  Nitrogen,  —  N  -f-  O2 ;  transparent, 
colorless  ;  wholly  irrespirable  ;  lighted  charcoal  and  phosphorus  burn 
in  it  with  increased  brilliancy. 

Olcfiant  gas — Bicarburctcd  hydrogen  gas — "  coal  gas," — C2  -j- 
H2 ;  transparent,  colorless,  tasteless,  nearly  inodorous,  when  pure  ; 
does  not  support  respiration  or  combustion  ;  a  lighted  taper  immersed 
in  it  is  immediately  extinguished.  It  burns  with  a  strong,  clear, 
white  light ;  mixed  with  oxygen,  in  the  proportion  of  1  volume  to  3, 
it  is  highly  explosive  and  dangerous. 

Phosphureted  hydrogen  gas,  —  P  -\-  H3 ;  colorless  ;  odor  highly 
offensive  ;  taste  bitter  ;  exists  in  the  vicinity  of  swamps,  marshes, 
and  grave -yards  ;  is  formed  by  the  decomposition  of  bones,  mainly  ; 
is  highly  inflammable  ;  takes  fire  spontaneously  on  coming  in  contact 
with  the  atmosphere  ;  mixed  with  pure  oxygen,  it  explodes.  It  is 
the  veritable  "  Will  o'  the  wisp." 

Sulphureted  hydrogen  gas —  Hydrosulphuric  acid  gas,  —  S  -j-  H  ; 
transparent,  colorless.;  taste  exceedingly  nauseous  ;  odor  offensive 
and  disgusting  ;  is  furnished  by  the  sulphurets  of  the  metals  in  gen- 
eral —  also  by  filthy  sewers  and  putrescent  eggs.  It  is  very  destruc- 
tive to  life  ;  placed  on  the  skin  of  animals,  it  proves  fatal.  It  burns 
with  a  pale  blue  flame,  and,  mixed  with  pure  oxygen,  it  is  explosive. 

Hydrocyanic  acid  —  Prussic  acid,  —  N  -j-  C2  -j-  H  ;  a  colorless, 
limpid,  highly  volatile  liquid;  odor  strong,  but  agreeable  —  similar 
to  that  of  peach-blossoms  ;  it  boils  at  79°  and  congeals  at  0  ;  exists 
in  laurel,  the  bitter  almond,  peach  and  peach  kernel.  It  is  a  most 
virulent  poison,  —  a  drop  placed  upon  a  man's  arm  caused  death  in  a 
few  minutes.  A  cat,  or  dog,  punctured  in  the  tongue  with  a  needle 
fresh  dipped  in  it,  is  almost  instantly  deprived  of  life.- 

Hydrofluoric  acid,  —  F  -j-  H  ;  a  colorless  liquid,  in  well  stopped 
lead  or  silver  bottles,  at  any  temperature  between  32°  and  59°.     It  is 


9i  PROPERTIES,    ETC.,    OP   BQDIES. 

obtained  by  tbe  action  of  sulphuric  acid  on  fluor-spar.  It  readily 
acts  upon  and  is  used  for  etching  on  glass.  It  is  the  most  destructive 
to  animal  matter  of  any  known  substance. 

Nitrohydrochhric  acid — "  aqua  regia,"  —  (1  part  nitric  acid  and  4 
parts  muriatic  acid,  by  measure  ;)  —  a  solvent  for  gold.  The  best  sol- 
vent for  gold  is  a  solution  of  sal  ammoniac  in  nitric  acid. 

Nitrosulphuric  acid, —  (1  part  nitric  acid  and  10  parts  sulphuric 
acid,  by  measure) — a  solvent  for  silver;  scarcely  acts  upon  gold, 
iron,  copper,  or  lead,  unless  diluted  with  water  ;  is  used  for  separat- 
ing the  silver  from  old  plated  ware,  &c.  The  best  solvent  for  silver, 
and  one  which  will  not  act  in  the  least  upon  gold,  copper,  iron,  or 
lead,  is  a  solution  of  1  part  of  nitre  in  10  parts  of  concentrated  sul- 
phuric acid,  by  weight,  heated  to  160°.  This  mixture  will  dissolve 
about  £  its  weight  of  silver.  The  silver  may  be  recovered  by  adding 
common  salt  to  the  solution,  and  the  chloride  decomposed  by  the  car- 
bonate of  soda. 

Selenic  acid,  —  Se  -j~  O3 ;  obtained  by  fusing  nitrate  of  potassa  with 
selenium  —  a  solvent  for  gold,  iron,  copper,  and  zinc. 

Silicic  acid,  —  (Silica  —  silex  ;  base  Silicon)  —  Si  -j-  O3 ;  exists 
largely  in  sand.  Common  glass  is  fused  sand  and  protoxide  of  potas- 
sium (carbonate  of  potassa  —  potash)  in  the  proportion  of  1  part  by 
weight  of  the  former  to  3  of  the  latter. 

Manganese,  compounded  with  oxygen,  in  different  proportions,  im- 
parts the  various  colors  and  tints  given  to  fancy  glass  ware,  now  se 
generally  in  vogue. 


SECTION  III. 
PRACTICAL  ARITHMETIC. 


VULGAR  FRACTIONS. 

A  fraction  is  one  or  more  parts  of  a  Unit. 

A  vulgar  fraction  consists  of  two  terms,  one  written  above  the 
other,  with  a  line  drawn  between  them. 

The  term  below  the  line  is  called  the  denominator,  as  showing  the 
denomination  of  the  fraction,  or  number  of  parts  into  which  the  unit 
is  broken. 

The  term  above  the  line  is  called  the  numerator,  as  numbering  the 
parts  employed.  These  together  constitute  the  fraction  and  its 
value. 

A  vulgar  fraction  always  denotes  division,  of  which  the  denomina- 
tor is  the  divisor  and  the  numerator  the  dividend.  Its  value  as  a  unit 
is  the  quotient  arising  therefrom. 

A  simple  fraction  is  either  a  proper  or  improper  fraction. 

A  proper  fraction  is  one  whose  numerator  is  less  than  its  denomina- 
tor, as  £,  f ,  |L,  &c. 

An  improper  fraction  has  its  numerator  equal  to  or  greater  than  its 
denominator,  as  f ,  £,  f  f ,  &c. 

A  mixed  fraction  is  a  compound  of  a  whole  number  and  a  fraction, 
as  lj,  6Ji,  12ft,  &c. 

A  compound  fraction  is  a  fraction  of  a  fraction,  as  £  of  |  ;  |  of 
4  of  if,  &c. 

A  complex  fraction  has  a  fraction  for  its  numerator  or  denom- 
inator, or  both,  as  |,  -,  |,  — ,  &c,  and  is  read  J  -J-  3  ;    4  -5-  g  ; 

£-H;  si-i-V&c. 

REDUCTION    OF   VULGAR    FRACTIONS. 
To  reduce  a  fraction  to  its  lowest  terms. 

This  consists  in  concentrating  the  expression  without  changing  the 
value  of  the  fraction  or  the  relation  of  its  parts. 

It  supposes  division,  and,  consequently,  by  a  measure  or  measures 
common  to  both  terms. 

It  is  said  to  be  accomplished  when  no  number  greater  than  1  will 
divide  both  terms  without  a  remainder:  —  therefore, 


96  VULGAR   FRACTIONS. 

Rule.  —  Divide  both  terms  by  any  number  that  will  divide  them 
without  a  remainder,  and  the  quotient  again  as  before  ;  continue  so  to 
do  until  no  number  greater  than  1  will  divide  them,  —  or  divioe  by 
the  greatest  common  measure  at  once. 

Example.  —  Reduce  jW?  l0  *ts  lowest  terms. 
*)-Afe-Wi^2-i|f  +  9-if-5-3-f     Ant. 

To  reduce  an  improper  fraction  to  a  mixed  or  whole  number. 

Rule.  —  Divide  the  numerator  by  the  denominator  and  to  the 
whole  number  in  the  quotient  annex  the  remainder,  if  any,  in  form  of 
a  fraction ,  making  the  divisor  the  denominator  as  before  ;  then  reduce 
the  fraction  to  its  lowest  terms. 

Example.     £«.  1±  ;  |J-  1T^  =  i£  ;  ff  -  2. 

To  reduce  a  mixed  fraction  to  an  equivalent  improper  fraction. 

Rule.  —  Multiply  the  whole  number  by  the  denominator  of  the 
fractional  part,  and  to  the  product  add  the  numerator,  and  place  their 
sum  over  the  said  denominator. 

Example.  —  Reduce  3£  and  12§  to  improper  fractions. 

3X4  =  12  4-1  =  ^-.     Ans.  12  X  9  +  8  —  ■!£&.     Ans. 

To  reduce  a  whole  number  to  an  equivalent  fraction  having  a  given 
denominator . 

Rule.  —  Multiply  the  whole  number  by  the  given  denominator, 
and  place  the  said  denominator  under  the  product. 

Example.  —  How  may  8  be  converted  into  a  fraction  whose  de- 
nominator is  12  ? 

8  X  12  —  f  f  •     Ans. 

To  reduce  a  compound  fraction  to  a  simple  one. 
Rule.  —  Multiply  all  the  numerators  together  for  a  numerator,  and 
all  the  denominators  together  for  a  denominator  ;  the  fraction  thus 
formed  will  be  an  equivalent,  but  often  not  in  its  lowest  terms.  Or, 
concentrate  the  expression,  when  practicable,  by  reciprocally  expung- 
ing, or  writing  out,  such  factors  as  exist  or  are  attainable  common  to 
both  terms,  and  then  multiply  the  remaining  terms  as  directed  above. 

Note.  —This  last  practice  is  called  cancellation,  or  canoeUloa  the  tafBI 
an  has  been  stated,  in  reciprocally  annulling,  or  casting  out,  aqua!  value-  from  both  terms, 
whereby  tho  expression  is  concentrated,  and  the  relation  of  the  parts  kept  umDaturbsd  ; 

ami  it  may  always  be  carried  to  tin:  extent  of  reducing  the  fraction  to  its  lowest  tonus, 
my  multiplication,  as  final,  is  resorted  to;  and  ofte.i.  therefore,  to  the  extent  that 
ruch  multiplication  is  inadmissible,  the  terms  having  been  cancelled  by  cueh  other  until 
but  a  single  numlier  is  left  in  each. 


VULGAR   FRACTIONS. 


97 


Example.  — Reduce  §  of  $  of  A  to  a  simple  fraction. 

Operation  by  multiplication,  |Xf  Xi53^33!-      Ans. 

23  1 
Operation  by  cancellation,   a       .  =  \.      Ans. 

Example.  —  Reduce  §  of  £  of  *g-  of  £  of  £  of  2    to  a   simple 
fraction. 

By  multiplication,  §  X  £  X  -V  X  f  X  $  X  \  -  Iff  2  -  f.    ^>». 
The  last  example  stated  )  2  3  12  6  5  2 
for  cancellation,  )  3  4    8    8  9 

PROCESS    OF    CANCELLING    THE    ABOVE. 

1.  The  3  in  num.  equals  the  3  in  denom.,  therefore  erase  both. 

2.  The  first  2  in  num.  equals  or  measures  the  4  in  denom.  twice,  therefore  place  a  2 
under  the  4,  and  erase  the  4  and  2  which  measured  it —  (as  4  :  2  :  :  2:1.) 

3.  The  2  (remaining  factor  of  4  and  2  erased)  in  denom.,  and  the  remaining  2  in  num., 
will  cancel  each  other,  —  erase  them. 

4.  The  12  and  6  in  num.  ==  72,  and  the  9  and  8  in  denom.  =72;  these,  therefore,  in 
their  relations  as  factors  equal  each  other,  and  may  be  erased. 

The  remaining  factors  represent  the  true  value  of  the  compound  fraction,  and  will  be 
found  =  |,  as  by  multiplication. 

Example.  —  Reduce  Tf  of  fa  to  a  simple  fraction. 

3 

0  3 

**  X  1.      Or  X$  X  ~  (=  18  +  6,  and  12  +  6)  =§  X  fa 


To  reduce  fractions  of  different  denominations  to  an  equivalent  simple 
one,  —  to  a  fraction  having  a  common  denominator. 
Rule.  — .  Multiply  each  numerator  by  all  the  denominators  except 
its  own  and  add  the  products  together  for  the  numerator,  and  multiply 
all  the  denominators  together  for  a  denominator. 

Note.  — Whole  numbers  and  fractions  other  than  simple,  must  first  be  reduced  to  sim- 
ple fractions  before  they  can  be  reduced  to  a  fraction  having  a  common  denominator. 

Example.  —  Reduce  f  and  |  to  an  equivalent  simple  fraction. 

2X3=8  +  9  =  J.£.        AnSt 

Example.  — Reduce  £,  f,  £,  and  *£  to  an  equivalent. 
^  +  f  =  T^  +  i  =  W  +  J/  =  Wir4-==6TyTT.     An*L 

9 


98  VULGAR   FRACTIONS. 

To  reduce  a  complex  fraction  to  a  simple  one. 
Rule  — Multiply  the  numerator  of  the  upper  fraction   by  the 
denominator  of  the  lower,  for  the  new  numerator  ;  and  the  denomi- 
nator of  the  upper  by  the  numerator  of  the  lower  for  the  new  denom- 
inator. 

1    4    1  51 

Examples.  —  Reduce  — ,  — ,  ^,  and  —  each  to  a  simple  fraction. 

l  +  l-i;  t+trVi  l  +  i-ixfc-fc-ii  si  = 

V,  and  V  X  J  =  V.  =  If-    **>■ 

To  reduce  Vulgar  Fractions  to  equivalent  Decimals. 

Rule.  — Divide  the  numerator  by  the  denominator  ;  the  quotient  is 
the  decimal,  or  the  whole  number  and  decimal,  as  the  case  may  be. 

Example.  —  Reduce  £,  4f ,  \ #,  to  decimals. 
7  4-8  =  0.875;  4|=-2£,=  4.6;   14  -J-  12  =  1.166  +.     Ans. 

To  find  the  greatest  common  measure  or  divisor  of  both  terms  of  a  simple 
fraction,  or  of  two  numbers. 

Rule.  —  Divide  the  greater  number  by  the  less ;  then  divide  the 
divisor  by  the  remainder  ;  and  so  on,  continuing  to  divide  the  last 
divisor  by  the  last  remainder  until  nothing  remains  ;  the  last  divisor 
is  the  greatest  common  measure  of  the  two  terms. 

Example.  —  What  is  the  greatest  common  measure  of  ££§  or  of 
132  and  256 1 

132  )  256  (  1 
132 

124  )  132  (  1 
124 

8 )  124  (  15 
120 

4)8(2 
8 


4.   Ans. 


To  find  the  least  common  denominator  of  two  or  more  fractions  of  dif- 
ferent denominators,  or  the  least  common  multiple  of  two  or  more 
numbers. 

Rule.  —  Divide  the  given  denominators,  or  numbers,  by  any  num- 
ber greater  than  1,  that  will  divide  at  least  two  of  them  without  a 
remainder,  which  quotient  together  with  the  undivided  numbers  set 
in  a  line  beneath.     Divide  the  second  line  as  before,  and  so  on,  until 


VULGAR   FRACTIONS.  99 

there  are  no  two  numbers  in  the  line  that  can  be  thus  divided  ;  the 
product  of  all  the  divisors  and  remaining  numbers  in  the  last  (undi- 
vided) dividend  is  the  ieast  common  denominator,  or  multiple  sought. 

Example.  —  What  is  the  least  common  denominator  of  2V  2^, 
and  •&,  or  of  20,  25,  and  50  ? 

5  )  20.25.50 

2)    4.5.10 

5  )    2.5.5 

2.1.1 

5X2X5X2=  100.     Arts. 

ADDITION    OF    VULGAR    FRACTIONS. 

Sum  of  the  products  of  each  numerator  with  all  the  denominators  except  that  of  th« 
numerator  involved,  forms  numerator  of  sum. 
Product  of  all  the  denominators  forms  denominator  of  sum. 

Rule.  —  Arrange  the  several  fractions  to  be  added,  one  after 
another,  in  a  line  from  left  to  right ;  then  multiply  the  numerator  of 
the  first  by  the  denominator  of  the  second,  and  the  denominator  of  the 
first  by  the  numerator  of  the  second,  and  add  the  two  products 
together  for  the  numerator  of  the  sum ;  then  multiply  the  two  denom- 
inators together  for  its  denominator  ;  bringdown  the  next  fraction,  and 
proceed  in  like  manner  as  before,  continuing  so  to  do  until  all  the 
fractions  have  been  brought  down  and  added.  Or,  reduce  all  to  a 
common  denominator,  then  add  the  numerators  together  for  the 
numerator  of  the  sum,  and  write  the  common  denominator  beneath. 

Examples.  —  Add  together  |,  §,  f,  and  ^. 

4X|  =  |X|  =  MX§  =  W-  =  -^=s3j.     Arts. 

i-*  +  i  =  *  =  tt.««i  !  +  ♦  =  *=»,  "dtf  +  tt-B 

=  -^  =  34.     Ans. 

SUBTRACTION  OF  VULGAR  FRACTIONS. 

Product  of  numerator  of  minuend  and  denominator  of  subtrahend,  forms  numerator  of 
minuend,  for  common  denominator. 

Product  of  numerator  of  subtrahend  and  denominator  of  minuend,  forms  numerator  of 
subtrahend,  for  common  denominator. 

Product  of  denominators  forms  common  denominator. 

Difference  of  new  found  numerators  forms  the  numerator,  and  common  denominator  the 
denominator,  of  the  difference,  or  remainder  sought. 

Rule.  —  Write  the  subtrahend  to  the  right  of  the  minuend,  with 
the  sign  ( — )  between  them  ;  then  multiply  the  numerator  of  the 
minuend  by  the  denominator  of  the  subtrahend,  and  the  denominator  of 
the  minuend  by  the  numerator  of  the  subtrahend ;  subtract  the  latter 
product  from  the  former,  and  to  the  remainder  or  difference  affix  tho 


100  VULGAR    FRACTIONS. 

product  of  the  two  denominators  for  a  denominator ',  the  sum  thus 
formed  is  the  answer,  or  true  difference. 

Examples.  —  Subtract  J  from  |,  also  f  from  \±. 

1-1  =  ^^  =  1  =  4-    -*«. 

if-i=55~51°8V     A**- 

DIVISION    OF    VULGAR    FRACTIONS. 

Product  of  numerators  of  dividend  and  denominators  of  divisor,  forms  numerator  of 
quotient. 

Product  of  denominators  of  dividend  and  numerators  of  divisor,  forms  denominator  of 
quotient;  llierefore, 

Rule.  —  Write  the  divisor  to  the  right  of  the  dividend  with  the 
sign  (-T-)  between  them ;  then  multiply  the  numerator  of  the  dividend 
by  the  denominator  of  the  divisor,  for  the  numerator  of  the  quotient, 
and  the  denominator  of  the  dividend  by  the  numerator  of  the  divisor, 
for  the  denominator  of  the  quotient.  Or,  invert  the  divisor,  and  mul- 
tiply as  in  multiplication  of  fractions.  Or,  proceed  by  cancellation, 
when  practicable. 

Examples.  —  Divide  \  by  j  ;  j  by  \  ;  ^  by  \\ ;  and  \  of  }  of 
|  of  ^  by  J  of  J  of  }  of  J. 

\*\w%\  l+i^ts  **4*-H5  *I5*HH-  Ans- 
1X1x1x1=^=^  and  jxixix§=F6ff-A. 

and  TV  +  -A  =  f|  -  ¥  =  6|.     Aw. 

FORM    FOR    CANCELLATION. EXAMPLE    LAST    GIVEN. 

1354         4243         20 

- — : — - — — -  =  — .     A?is.,  as  above. 

2     4     6     3         113     2  3  ' 

Note.  —  The  foregoing  example  can  be  cancelled  to  the  extent  of  leaving  but  a  A  and  a 
5  (=20)  numerators,  and  a  3  denominator.  Units,  or  l's,  in  the  expressions,  are  value- 
less, as  a  sum  multiplied  by  1  is  not  increased. 

MULTIPLICATION    OF    VULGAR    FRACTIONS. 

Product  of  numerators  of  multiplier  and  multiplicand,  forms  numerator  of  product. 
Product  of  denominators  of  multiplier  and  multiplicand,  forms  denominator  of  product. 

Rule.  —  Multiply  the  numerators  together  for  a  numerator,  and  the 
denominators  together  for  the  denominator. 

I  x  amples.  —  Multiply  \  by  \  ;  J  by  7  ;  JJ  by  ^  ;  \  of  \  of  } 

Of  I  Off. 

I'XfHh  3xi  =  v  ;  px-p-WV-f  *x§xf 

-A-i.an«llX|Xi  =  A  =  i,andiXi=»TV      Ans 


VTTLGATt    tRACTiGNS.  1Q1 


It  has  been  seen  that  a  compound  fraction  is  converted  into  an 
equivalent  simple  one,  by  multiplying  the  numerators  together  for  a 
numerator,  and  the  denominators  together  for  a  denominator;  and 
it  has  also  been  seen  that  a  series  of  simple  fractions  are  con- 
verted into  a  product,  by  the  same  process.  It  is  therefore  evident 
that  compound  fractions  and  simple,  or  a  series  of  compound  and  a 
series  of  simple,  may  be  multiplied  into  each  other,  for  a  product,  by 
multiplying  all  the  numerators  of  both  together  for  a  numerator,  and 
all  the  denominators  of  both  together  for  a  denominator;  and  that  the 
product  will  be  the  same  as  would  be  obtained,  if  the  compound  were 
first  converted  into  an  equivalent  simple  fraction,  and  the  simple  frac- 
tions into  a  product  or  factor,  and  these  multiplied  together  for  a 
product. 

It  has  also  been  seen  that  a  fraction  is  divided  by  a  fraction  by  mul- 
tiplying the  numerator  of  the  dividend  by  the  denominator  of  the 
divisor,  for  the  numerator  of  the  quotient,  and  the  denominator  of  the 
dividend  by  the  numerator  of  the  divisor,  for  the  denominator  of  the 
quotient ;  and  that  this  multiplication  becomes  direct  as  in  multiply- 
ing for  a  product,  if  the  divisor  is  inverted.  And  it  is  clear  that  a 
compound  divisor,  or  a  series  of  simple  divisors,  or  both,  may  be  used 
instead  of  their  simple  equivalent,  and  with  the  same  result,  if  all  are 
inverted. 

It  is  therefore  evident  that  any  proposition,  or  problem,  in  fractions, 
consisting  of  multiplications  and  divisions  both,  and  these  only,  no 
matter  how  extensive  and  numerous,  or  whether  in  compound  frac- 
tions, or  simple,  or  both,  may  be  solved,  and  the  true  result  obtained, 
as  a  product,  by  simply  multiplying  all  the  numerators  in  the  state- 
ment together  for  a  numerator,  and  all  the  denominators  in  the  state- 
ment for  a  denominator,  all  the  divisors  in  the  statement  being 
inverted  ;  that  is,  all  the  numerators  of  the  divisors  being  made  denom- 
inators in  the  statement,  and  all  the  denominators  of  the  divisor  being 
made  numerators  in  the  statement.  And  it  is  further  evident  that  a 
proposition  stated  in  this  way,  admits  of  easy  cancellation  as  far  as 
cancellation  is  practicable,  which  is  often  to  great  extent. 

Example.  — It  is  required  to  divide  12  by  |  of  | ;  to  multiply  the 
quotient  by  the  product  of  4  and  8 ;  to  divide  that  product  by  £  of  | 
of  8  ;  to  multiply  the  quotient  by  £  of  ■§  of  T9^ ;  and  to  divide  that 
product  by  the  product  of  5  and  9. 

9* 


102  VULGAR   FRACTIONS. 

8TATEMENT. 
(Dividends  read  from  right  to  left,  divisors  from  left  to  right) 


Numerators  of  dividends  and  denominators  of  divisors. 

i  &    §     i      i     i 

3       O  3  <0  3  O 

Numerator  of?        7^£^£~£^£^£  S  Dividend  of 

statement.       S    •-<  ( statement. 


Denominator  )         ««  l^rtaooocorjjiOda     <  Divisor  of 

of  statement.  \        ^^-^  ^  — ^..^  ^_,  i  statement. 

jo  jo  t^f  n 

•aioeiAipjo  BjotBaauinu  pus  spuaptAtp  jo  r-JoiBUtujouaa 


The  answer  to  the  above  proposition  is  1£|,  and  the  proposition 
as  stated  may  be  readily  cancelled  to  its  lowest  terms.  It.  may  be 
cancelled  to  the  extent  of  leaving  but  4,  4,  2  in  the  numerator,  and  7, 
3,  in  the  denominator,  ^J^-2-  =  §f  —  l^f- 

To  reduce  a  fraction  in  a  higher  denomination  to  an  equivalent  fraction 
in  a  given  lower  denomination. 
Rule.  —  Multiply  the  fraction  to  be  reduced — numerators  into 
numerator  and  denominators  into  denominator  —  by  a  fraction  whose 
numerator  represents  the  number  of  parts  of  the  lower  denomination, 
required  to  make  one  of  the  denomination  to  be  reduced. 

Example.  —  Reduce  I  of  a  foot  to  an  equivalent  fraction  in  inches. 

Example.  —  Reduce  |  of  a  pound  to  an  equivalent  fraction  in  § 
ounces. 

$  X  Jf  -  -85°-  +  I  =  W  "  ¥•     Ans' 
Or)fXJr6-Xf  =  ^0-==-2f     Arts. 

To  reduce  a  fraction  in  a  lower  denomination  to  an  equivalent  fraction 
in  a  gwen  higher  it  nrnni  nation. 
Rule. — Multiply  the  fraction  to  be  reduced  —  numerator  into 
denominator  and  denominator  into  numerator  —  by  a  fraction  whose 
numerator  represents  the  number  of  parts  required  of  the  lower 
denomination  to  make  1  of  the  higher. 

•mple.  —  Reduce  Q  inches  to  an  equivalent  fraction  in  feet. 
%l  .*- J£  «.  jj  «■  J.    Ans.       Or,  V  X  TV -=  §*  ~  £•     Ans. 


*  VULGAR  FRACTIONS.  103 

Example.  —  Reduce  *g-  two  third  ounces  to  an  equivalent  frac- 
tion iir  pounds. 

¥X§  =  -8^-7-JT6-  =  fS-t.     Ans. 
Or,  VX§XtV -*»-"*■     Ans. 

To  reduce  a  fraction  in  a  higher  to  whole  numbers  in  lower  denomi- 
nations,. 
Rule.  —  Multiply  the  numerator  of  the  given  fraction  by  the  num- 
ber of  parts  of  the  next  lower  denomination  that  make  one  of  the 
given  fraction,  and  divide  the  product  by  the  denominator.  Multiply 
the  numerator  of  the  fractional  part  of  the  quotient  thus  obtained  by 
the  number  of  parts  in  the  next  lower  denomination  that  make  1  of 
the  denomination  of  the  quotient,  and  divide  by  its  denominator  for 
whole  numbers  as  before  ;  so  proceed  until  the  whole  numbers  in  each 
denomination  desired  are  obtained. 

Example. — How  many  hours,  minutes,  and  seconds,  in  y9^  of  a 
day? 

^24  =  2^6   =  ^^  |X60=  ±^Q_  =  25,  $X  6  0  -  3  OQ.  „  42  ^  . 

15  h.,  25  m.,  42f-  sec.     Ans. 
Example. — How  many  minutes  in  T9¥  of  a  day  ? 

_S.  X  2 4  X  6  0  =±2_9_60  =  925^       Ans, 

To  reduce  fractions,  or  ichole  numbers  and  fractions,  in  loioer  denomi- 
nations, to  their  value  in  a  higher  denomination. 
Rule.  —  Reduce  the  mixed  numbers  to  improper  fractions,  find 
their  common  denominator,  and  change  each  whole  number  and 
numerator  to  correspond  therewith.  Then  reduce  the  higher  numbers 
to  their  values  in  the  lowest  denomination,  add  the  value  in  the  lowest 
denomination  thereto,  and  take  their  sum  for  a  numerator.  Multiply 
the  common  denominator  by  the  number  required  of  the  lowest  denom- 
ination to  make  one  of  the  next  higher,  that  product  by  the  number 
required  of  that  denomination  to  make  1  of  the  next  higher,  and  so 
on,  until  the  highest  denomination  desired  is  reached,  and  take  the 
product  for  a  denominator,  and  reduce  to  lowest  terms. 

Example.  —  Reduce  5\  oz.,  3^  dwts.,  2^  grs.,  troy,  to  lbs. 
Y-  •  ¥"  •  J*-1**^"  ;  therefore, 
160  X  20  =  3200 
96 

3296  X  24  =  79104 

75 


79179 


30  X  24  X  20  X  12  =  172800 


=  .458 -fibs.   Ans. 


104  DECIMAL   FRACTIONS. 

Example.  —  Reduce  11  hours,  59  minutes,  60  seconds,  to  the  frac- 
tion of  a  day. 

11  X  60  =  660 
59 

719  X  60  =  43140 
60 


43200  . 

Ans. 


60  X  60  X  24  =  86400 


!-'■ 


Example.  —  Reduce  15  h.,  25  m.,  42f  sec,  to  the  fraction  o\  a 
day. 

15  X  60  X  60  =  54000 
25  X  60  =    1500 
42f 

55542f 

7 


388800 


7  X  00  X  60  X  24  =  604800 

To  work  fractions,  or  whole  numbers  and  fractions,  by  the  Rule  of 
Three,  or  Proportion. 

Rule.  —  Reduce  the  mixed  terms  to  simple  fractions,  state  the 
question  as  in  whole  numbers,  invert  the  divisor,  and  multiply  and 
divide  as  in  whole  numbers. 

Example.  —  If  2£  yards  of  cassimere  cost  $4J,  what  will  |  of  a 
"ard  cost?     2 J  =  £  ;  4 J  =  Y"  »  then> 

*:■¥■••-•  I^^Y-SfSf-W-Si^.      Ans. 


DECIMAL  FRACTIONS. 

A  decimal  fraction  is  written  with  its  numerator  only.  Its  denomi- 
nator is  understood.  It  occupies  one  or  more  places  of  figures,  and 
has  a  point  or  dot  (.)  prefixed  or  placed  before  it.  The  dot  (.)  alone 
distinguishes  it  from  an  integer  or  whole  number.  It  supposes  a 
denominator  whose  value  is  a  unit  broken  into  parts,  having  a  ten- 
fold relation  to  the  number  of  places  the  numerator  occupies.  The 
denominator,  therefore,  of  any  decimal  is  always  a  unit  (1)  with  as 
many  ciphers  annexed  as  the  numerator  has  places  of  figures.  Thus, 
the  denominator  of  .1,  .2,  .3,  &c,  is  10,  and  the  fractions  are  read, 
one  tenth,  two  tenths,  three  tenths,  &c.  The  denominator  of  .01,  .11, 
.12,  &c,  is  100,  and  these  are  read,  one  hundredth,  eleven  hundredths, 


DECIMAL   FRACTIONS.  105 

twelve  hundredths,  &c.  The  denominator  of  .001,  .101,  .125,  &c,  is 
1000,  and  these  are  read  one  thousandth,  one  hundred  and  one  thousandths, 
one  hundred  and  twenty-Jive  thousandths,  &c.  The  denominator  of  a 
decimal  occupying  four  places  of  figures  as  .7525  is  10000,  and  so  on 
continually. 

The  first  figure  on  the  right  of  the  decimal  point  is  in  the  place  of 
tenths,  the  second  in  the  place  of  tenths  of  tenths,  or  hundredths,  the 
third  in  the  place  of  tenths  of  tenths  of  tenths,  or  thousandths,  &c. 
Thus  the  value  of  a  decimal  occupying  four  places  of  figures,  as 

»«**  ,       ,   •  7525     752*    75.*     lh  ± 

•7525,  for  example,  is  ,  « ,  =  ,  =  — ±   -4 —  «= 

__l__  r     10000     1000    100     10  ~  100 

£  i 

1 .       A  decimal  is  converted   into  a  vulgar  fraction  of 

1      '    100  ^ 

equal  value,  by  affixing  its  denominator. 

Ciphers  placed  on  the  right  of  decimals  do  not  change  their  value. 
Thus,  .1850  =  .185,  plainly  for  the  reason  that  the  denominator  of 
the  latter  bears  the  same  relation  to  that  of  the  former  that  185  bears 
to  1850  ;  from  both  terms  of  the  fraction  a  ten  fold  has  been  dropped. 

Ciphers  placed  on  the  left  of  decimals  decrease  their  value  ten  fold 
for  every  cipher   so  placed.     Thus,  .1  =  -jVj,   «01  =  T£T,  .001  = 

A  mixed  number  is  a  whole  number  and  a  decimal.  Thus,  4.25  is 
a  mixed  number.  Its  value  is  4  units,  or  ones,  and  -^Jfr  of  1,  = 
^-^  =  4£.  The  number  on  the  left  of  the  separatrix  is  always  a 
whole  number  —  that  on  its  right,  always  a  decimal. 

ADDITION    OF    DECIMALS. 

Rule.  —  Set  the  numbers  directly  under  each  other  according  to 
their  values,  whole  numbers  under  whole  numbers,  and  decimals  un- 
der decimals  ;  add  as  in  whole  numbers,  and  point  off  as  many  places 
for  decimals  in  the  sum  as  there  are  figures  in  that  decimal  occupying" 
the  greatest  number  of  places. 

Examples.  —  Add  together  .125,  .34,  .1,  .8672.  Also,  125,  34.11, 
.235.  1.4322. 


.125 

.34 

.1 

.8072 
1.4322     Ans. 


125. 
34.11 
.235 
1.4322 
i1«l7772     Ans. 


SUBTRACTION    OF    DECIMALS. 

Rule.  —  Set  the  numbers,  the  less  under  the  greater,  and  in  other 
icsoects  as  directed  for  addition ;  subtract  as  in  whole  numbers,  and 


106 


DECIMAL   FRACTIONS. 


point  off  as  many  places  for  decimals  in  the  remainder  as  the  decimal 
having  the  greatest  number  of  figures  occupies  places. 


Examples.  — 

.8 
.2653 


Subtract  .2653  from  .8. 


.5347     Arts. 


Also,  11.5  from  238.134. 

238.134 

11.5 
226.634     Ans. 


MULTIPLICATION    OF    DECIMALS. 
Rule.  —  Multiply  as  in  whole  numbers,  and  point  off  as  many 
places  for  decimals  in  the  product  as  there  aTe  decimal  places  in  the 
multiplicand  and  multiplier  both.     If  the  product  has  not  so  many 
places,  prefix  ciphers  to  supply  the  deficiency. 

Examples.  —  Multiply  14.125  by  3.4.     Also,  5.14  by  .007. 


14.125 
3.4 
56500 
42375 
48.0250  =  48.025.    Ans. 


5.14 
.007 


.03598    Arts. 


Notb.  —  Multiplying  by  a  decimal  is  equivalent  to  dividing  by  a  whole  number  that 
hears  the  same  relation  to  a  unit  that  a  unit  bears  to  a  decimal.  Multiplying  by  a  deci- 
mal, therefore,  is  equivalent  to  dividing  by  the  denominator  of  a  fraction  of  equal  value; 
whose  numerator  is  1,  or  of  dividing  by  the  denominator  of  a  fraction  of  equal  value  whose 
numerator  is  more  than  1,  and  multiplying  the  quotient  by  the  numerator.  Thus,  the 
decimal  .23  ==  ^j  =  {,  and  the  decimal  .875  =  yV^  =  J.  And  14.23  X  -2"'  — 
8.6676.  and  14.23  -j-  4  =a  3.5575.  So,  also,  1 1.23  X  -875  =  12.45125,  and  14.23  -f-  8  = 
X  7  =  12  1.")  125.  It  is  sometimes  a  saving  of  labor  and  matter  of  convenience  to 
achieve  multiplication  by  this  process. 

DIVISION  OF  DECIMALS. 
Rule.  —  Write  the  numbers  as  for  division  of  whole  numbers,  then 
remove  the  separatrix  in  the  dividend  as  many  places  of  figures  to  the 
right,  (supplying  the  places  with  ciphers  if  they  are  not  occupied,)  as 
there  are  decimal  figures  in  the  divisor ;  consider  the  divisor  a  whole 
number  and  divide  as  in  division  of  whole  numbers. 


Example.  —  Divide  .5  by  .17.     Also,  .129  by  4. 


.17). 50(2.94+.     Ans. 
34 

160 
153 
70 


4).129(.032-f-. 
12 

9 

I 


Ans. 


DECIMAL    FRACTIONS. 


107 


Examples.  — Divide  16.5  by  1.232.  Also,  1.2145  by  12.231. 

12.231,) l.214,50(.099294- >  A 

1  100  79  .0993—  f  Ans' 


.232,}  16.500,  (13.3928-L-.  Ans 
1232 


4180 

3696 
4840 
3696 
11440 
11088 


113  710 
110  079 
3  6310 
2  4462 


1  18480 
1  10079 
8401 


3520 
2464 
10560 
,9856 
704 

Notk.  —  Dividing  by  a  decimal  is  equivalent  to  multiplying  by  a  whole  number  that 
bears  the  same  proportion  to  a  unit  that  a  unit  bears  to  the  decimal.  Dividing  by  a  deci- 
mal, therefore,  is  equivalent  to  multiplying  by  the  denominator  of  a  fraction  of  equal 
value  whose  numerator  is  1,  or  multiplying  by  the  denominator  of  a  fraction  of  equal 
value  whose  numerator  is  more  than  1,  and  dividing  the  product  by  the  numerator.  Di- 
viding by  a  fraction  is  equivalent  to  multiplying  by  its  denominator  and  dividing  the 
product  by  its  numerator,  or  dividing  by  its  numerator  and  multiplying  the  quotient  by 
its  denominator.  Thus,  .5  =  T50  as  \,  and  .75  =  ^fo  =  J.  And  12.24  -f-  .5  =  24.43, 
and  12.24  V  2  =  24. 13.  So,  also.  12.24  -i-  .75  =  16.  32,  and  12.24  X  4  =  4S-96  +  3  =* 
16.32.    This  method  of  accomplishing  division  may  often  be  resorted  to  with  convenience. 

REDUCTION    OF    DECIMALS. 

To  reduce  a  decimal  in  a  higher  to  whole  members  in  successive  lower 
denominations. 
Rule. — Multiply  the  decimal  by  that  number  in  the  next  lower 
denomination  that  equals  one  of  the  denomination  of  the  decimal,  and 
point  off  as  many  places  for  a  remainder  as  the  decimal  so  multiplied 
has  places.  Multiply  the  remainder  by  the  number  in  the  next  lower 
denomination  that  equals  1  of  the  denomination  of  the  remainder,  and 
point  off  as  before  ;  so  continue,  until  the  reduction  is  carried  to  the 
lowest  denomination  required. 

Example.  — What  is  the  value  of  .62525  of  a  dollar? 
.       .62525 

100 


Cents,      62.52500 
10 

Mills, 


5.25000        An..     62  cents  5|  mills. 


108  DECIMAL   FE ACTIONS. 

Example.  —  What  is  the  value  of  .46325  of  a  barrel? 

.46325 
32 


Gallons,  14.82400 
4 


Quarts.       3.296 


Pints,  .592 

4 


Gills,         2.368.     Ans.      14  gals.  3  qts.  2^6^  gilfc. 

Example.  —  How  many  pence  in  .875  of  a  pound  ? 
'  .875  X  240  -»  210.     Ans. 

To  reduce  decimals,  or  whole  numbers  and  decimals,  in  lower  denomin- 
ations, to  their  value  in  a  higher  denomination. 
Rule.  —  Reduce  all  the  given  denominations  to  their  value  in  the 
lowest  denomination,  then  divide  their  sum  by  the  number  required  of 
the  lowest  denomination  to  make  one  of  the  denomination  to  which 
the  whole  is  to  be  reduced. 

Example.  —  Reduce  14  gallons,  3  quarts,  2.368  gills,  to  the  deci- 
mal of  a  barrel. 

14  X  4  =  56  -f-  3  =  59  X  8  =  472  -f  2.368  =  474.368. 
8  X  4  X  32  =  1024  )  474.368  (  .46325.     Ans. 

To  work  decimals,  or  whole  numbers  and  decimals,  by  the  Rule  of 
Three,  or  Proportion. 
Rule.  —  State  the  question  and  work  it  as  in  whole  numbers, 
taking  care  to  point  off  as  many  places  for  decimals  in  the  product  to 
be  used  as  the  dividend,  as  there  are  decimals  in  the  two  terms  which 
form  it,  and  to  remove  the  decimal  point  therein  as  many  places  to  the 
right  as  there  are  decimals  in  the  term  to  be  used  as  a  divisor,  before 
the  division  is  had. 

Example.  —  If  .75  of  a  pound  of  copper  is  worth  .31  of  a  dollar 
how  much  is  3.75  lbs.  worth  ? 

.75  :  .31  ::  3.75 
.31 
375 
1125 


.75)  1.16,25  ($1.55.     Ans. 


PROPORTION.  X09 

PROPORTION,  OR  RULE  OF  THREE. 

The  Rule  of  Proportion  involves  the  employment  of  three  terms 
—  a  divisor  and  two  factors  for  forming  a  dividend — and  seeks  a 
quotient,  which,  when  the  proposition  is  written  in  ratio,  bears  the 
same  relation  to  the  third  term  that  the  second  term  bears  to  the  first 
Two  of  the  terms  given  are  of  like  name  or  nature,  and  the  other  is 
of  the  name  or  nature  of  the  quotient  or  answer  s«ught.  That  of 
the  nature  of  the  answer  is  always  one  of  the  factors  for  forming  the 
dividend,  and,  if  the  answer  is  to  be  greater  than  that  term,  the  larger 
of  the  remaining  two  is  the  other ;  but  if  the  answer  is  to  be  less 
than  that  term,  the  less  of  the  remaining  two  is  the  other — the 
remaining  term  is  the  divisor. 

Example.  —  If  $12  buy  4  yards  of  cloth,  how  many  yards  will 
$108  buy? 

£  X  108  108 

jg—  =  —  m  36  yards.     Ans. 

3 

Example.  — If  4  yards  of  cloth  cost  $12,  how  many  dollars  will 
36  yards  cost? 

11^l.—  =  108  dollars.     Ans. 
4 

Example.  —  If  30  men  can  finish  a  piece  of  work  in  12  days,  how 
many  men  will  be  required  to  finish  it  in  8  days  ? 

30  X  12  , 

—  =  45  men.     Ans. 

8 

Example. — If  45  men  require  8  days  to  finish  a  piece  of  work, 
how  many  men  will  finish  the  same  work  in  12  days? 

— — —  =  30rnen.     Ans. 
IS 

Example.  —  If  8  days  are  required  by  45  men  to  finish  a  piece 
of  work,  how  many  days  will  be  required  by  30  men  to  finish  the 
same  work? 

8  X  45 

30      =  12  days.     Ans. 

Example.  — If  12  days  are  required  by  30  men  to  perform  a  piece 
of  work,  how  many  days  will  be  required  by  45  men  to  do  the  same 
work? 

12  X  30 

— — —  =  8  days.     Ans. 
45 

Example.  —  I  borrowed  of  my  friend  $150,  which  I  kept  3  months, 
and,  on  returning  it,  lent  him  $200  ;  how  long  may  he  keep  the  sum 
10 


110  COMPOUND    PROPORTION. 

that  the  interest,  at  the  same  rate  peT  cent.,  may  amount  to  that  which 
his  own  would  have  drawn ! 

150  X  3  -J-  200  =  2£  months.     Ans. 

Example.  —  A  garrison  of  250  men  is  provided  with  provisions  for 
30  days,  how  many  men  must  be  sent  out  that  the  provisions  may  last 
those  remaining  42  days? 

250  X  30  -j-  42  =  179,  and  250  —  179  =  71.     Ans. 

Example.  — If  to  the  short  arm  of  a  lever  2  inches  from  the  ful- 
crum there  be  suspended  a  weight  of  100  lbs.,  what  power  on  the 
long  arm  of  the  lever  20  inches  from  the  fulcrum  will  be  required  to 
raise  it  1 

20  :  2  ::  100=  10  lbs.     Ans. 

Example.  —  At  what  distance  from  the  fnlcrum  on  the  long  arm  of 
a  lever  must  I  place  a  pound  weight,  to  equipoise  or  weigh  20  lbs., 
suspended  2  inches  from  the  fulcrum  at  the  other  end  1 
1  :  2  ::  20  :  40  inches.     Ans. 

Note.  —  If  we  examine  the  foregoing  with  reference  to  the  fact,  we  shall  see  that  every 
proposition  in  simple  proportion  consists  of  &4erm  and  a  half!  or,  in  other  words,  of  a 
compound  term  consisting  of  two  factors,  and  a  factor  for  which  another  factor  is  sought 
that  together  shall  equal  the  compound.  We  have  only  to  multiply  the  factors  of  the 
compound  together  —  and  a  little  observation  will  enable  us  to  distinguish  it  — and  divide 
by  the  remaining  factor,  and  the  work  is  accomplished.    See  Compound  Proportion. 


COMPOUND  PROPORTION,  OR  DOUBLE  RULE  OF  THREE. 

Compound  Proportion,  like  single  proportion,  consists  of  three 
terms  given  by  which  to  find  a  fourth — a  divisor  and  two  factors  for 
forming  a  dividend  —  but  unlike  single  proportion,  one  or  more  of  the 
terms  is  a  compound,  or  consists  of  two  or  more  factors ;  #nd  some- 
times a  portion  of  the  fourth  term  is  given,  which,  however,  is  always 
a  part  of  the  divisor. 

Of  the  given  terms,  two  are  suppositive,  dissimilar  in  their  natures, 
and  relate  to  each  other,  and  to  each  other  only ;  and  upon  their  rela- 
tion'the  whole  is  made  to  depend  ;  the  remaining  term  is  of  the  nature 
of  one  of  the  former,  and  relates  to  the  fourth  term,  which  is  of  tho 
nature  of  the  other. 

The  object  sought  is  a  number,  which,  multiplied  into  the  factor  or 
factors  of  the  fourth  term  given,  if  any,  and  if  not,  which  of  itself, 
beam  the  same  proportion  to  the  dissimilar  term  to  which  it  relates, 
as  the  suppositive  term  of  like  nature  hears  to  the  term  to  which  it 
relates. 

Rule. — Observe  the  denomination  in  which  the  demand  is  mad.1, 
and  of  the  suppositive  terms  make  that  of  like  nature  the  second,  and 
the  other  the  first ;  maVe  the  remaining  term  the  third  term  ;  and,  if 


COMPOTTND    PROPORTION.  Ill 

there  are  any  factors  pertaining  to  the  fourth  term,  ufllx.  them  to  the 
first ;  multiply  the  second  and  third  terms  together  and  divide  hy  the 
first,  and  the  quotient  is  the  answer,  term,  or  portion  of  a  term, 
sought. 

Example.  —  If  12  horses  in  6  days  consume  36  bushels  of  oats, 
how  many  bushels  will  suffice  21  horses  7  days* 
12  x  6  :  36  ::2l  x  7  :  *• 
3 

30  X  21  X  7  147 

7~i ^ —  =  — —  =  73£  bushels.     Ans. 

&%  X  v  2 

2 

Example.  —  If  12  horses  in  6  days  consume  36  bushels  of  oata, 
how  many  horses  will  consume  73£  bushels  in  7  days  ? 
36  :  12  x  6  ::  73£  :  7  x  *• 
12  X  6  X  73£        147 
36    X    7      ==  T  =  21  h0TSeS-     AnS' 

Example. — If  the  interest  on  $1  is  1.4  cts.  for  73  days,  (exact 
interest  at  7  per  cent.,)  what  will  be  the  interest  on  $150.42  for  146 
days? 

73  :  1.4  ::  150.42  X  146  :  x. 
1.4  X  180.48  X  146  ^  ^     ^ 

Example. — If  the  interest  on  $1  is  1.2  cts.  for  73  days,  (exact 
interest  at  6  per  cent.,)  what  will  be  the  interest  on  $125  for  90 
days? 

73  :  1.2  ::  125  X  90  :  x  —  $1.85.     Ans. 

Example.  —  If  $100  at  7  per  cent,  gain  $1.75  in  3  months,  how 
much  at  6  per  cent,  will  $170  gain  in  11£  months? 

100  x  7  X  3  :  1.75  ::  170  x  6  x  11.5  '  *. 
1.75  X  170  X  6  X  11-5  -j-  100  X  7  X  3  =  $9.77,5.     Ans. 

Example.  —  By  working  10  hours  a  day  6  men  laid  22  rods  of  wall 
in  3  days  ;  how  many  men  at  that  rate,  who  work  but  9  hours  a  day, 
will  lay  40  rods  of  wall  in  8  days  ? 

22  :  6  x  3  X  10  ::  40  :  9  x  8  X  -r- 

6  X  3  X  10  X  40  -r  22  X  9  X  8  =  4T6T.      Ans. 

Example. — If  it  costs  $112  to  keep  16  horses  30  days,  and  it 
costs  as  much  to  keep  2  horses  as  it  costs  to  keep  5  oxen,  how  much 
will  it  cost  to  keep  28  oxen  36  days  ? 


112  CONJOINED   PROPORTION,  OK   CHAIN   RTTLE, 

•  16  x  30  :  112  : :  f  x  28  x  26  :  x. 

Or,—  16  X  30  X  5  :  112  ::  28  X  36  X  2  :  ar. 

Ill    28    ff*^  88X18  XT 
U    30      5  5X5 

5 

Example. — If  24  men,  in  8  days  of  10  hours  each,  ean  dig  a 
trench  250  feet  long,  8  feet  wide,  and  4  feet  deep,  how  many  men,  in 
12  days  of  eight  hours  each,  will  be  required  to  dig  a  trench  80  feet 
long,  6  feet  wide,  and  4  feet  deep  ? 
250X8X4:24X8X10  -  80X6X4  :  12X8X^=5— .    Ans. 

Example.  — If  120  men  in  six  months  perform  a  given  task,  work- 
ing 10  hours  a  day,  how  many  men  will  be  required  to  accomplish  a 
like  task  in  5  months,  working  9  hours  a  day? 

120  X  6  X  10  =  5  X  9  X  x. 
Or,  —  I  :  120  X  6  x  10  ::  1  :  5  X  9  X  ^.  ==  160.    Ans. 

Example.  — The  weight  of  a  bar  of  wrought  iron,  1  foot  in  length, 
1  inch  in  breadth,  and  1  inch  thick,  being  3.38  lbs.,  (and  it  is  so,) 
what  will  be  the  weight  of  that  bar  whose  length  is  12£  feet,  breadth 
3|  inches,  and  thickness  |  of  an  inch  ? 

l  :3.38  ::  12.5  X  3.25  x  .75  :  x. 
Or,  —  1  :  3.38  ::  3£  X  *&  X  f  : »,  and 
3.38X25X13X3  =  1  lbg 

2X4X4  ^ 

Example.  —  The  weight  of  a  bar  of  wrought  iron,  one  foot  in  length 
and  1  inch  square,  being  3.38  lbs.,  what  length  shall  I  cut  from  a  bar 
whose  breadth  is  2f  inches,  and  thickness  £  inch,  in  order  to  obtain 

10  lbs.?     3.38  :  l  "  10  :  -y.  x  £  X  *• 

1  X  10  X  4  X  2 

=  2  feet  l^Pn  inches.     Ans. 

3.38  X  11  X  1 


CONJOINED  PROPORTION,  OR  CHAIN  RULE. 

The  Chain  Rule  is  a  process  for  determining  the  value  of  a  given 
quantity  in  one  denomination  of  value,  in  some  other  given  denomi- 
nation of  value  ;  or  the  immediate  relationship  which  exists  between 
two  denominations  of  value,  by  means  of  a  chain  of  approximate  steps, 


CONJOINED   PROPORTION,    OR   CHAIN   RULE.  113 

circumstances,  or  equivalent  values,  known  to  exist,  which  connect 
them.  In  every  instance  at  least  five  terms  or  values  are  employed 
in  the  process,  and  in  all  instances  the  number  employed  will  be  un- 
even. A  proposition  involving  but  three  terms,  of  this  nature,  is  a 
question  in  single  proportion.  The  equivalent  values  employed  are 
divided  into  antecedents  and  consequents,  or  causes  and  effects  ;  and  the 
value  or  quantity  for  which  an  equivalent  is  sought,  is  called  the  odd 
term. 

Rule.  —  1.  When  the  value  in  the  denomination  of  the  first  antece- 
dent is  sought  of  a  given  quantity  in  the  denomination  of  the  last  conse- 
quei\t. —  Multiply  all  the  antecedents  and  the  odd  term  together  for  a 
dividend,  and  all  the  consequents  together  for  a  divisor;  the  quotient 
will  be  the  answer  or  equivalent  sought. 

Rule.  —  2.  When  the  value  in  the  denomination  of  the  last  consequent 
is  sought  of  a  given  quantity  in  the  denomination  of  the  first  antecedent, 
—  Multiply  all  the  consequents  and  the  odd  term  together  for  a  divi- 
dend, and  all  the  antecedents  together  for  a  divisor ;  the  quotient  will 
be  the  answer  required. 

Example.  —  I  am  required  to  give  the  value,  in  Federal  money,  of 
5  Canada  shillings,  and  know  no  immediate  connection  or  relationship 
between  the  two  currencies  —  that  of  Canada  and  that  of  the  United 
States.  The  nearest  that  I  do  know  is  that  20  Canada  shillings  have 
a  value  equal  to  32  New  York  shillings,  and  that  12  New  York  shil- 
lings equal  in  value  9  New  England  shillings,  and  that  15  New  Eng- 
land shillings  equal  $2.50  ;  and  with  this  knowledge  will  seek  the 
value,  in  Federal  money,  of  the  5  Canada  shillings. 
2^0X^X32X5  Am_ 

15  X  12  X  20 

Example.  —  If  $2£  equal  15  New  England  shillings,  and  nine  shil- 
lings in  New  England  equal  12  shillings  in  New  York,  and  32  shil- 
lings in  New  York  equal  20  shillings  in  Canada,  how  many  shillings 
in  Canada  will  equal  $1 1 

*4£U^ » « 

3     * 
Example.  —  If  14  bushels  of  wheat  weigh  as  much  as  15  bushels 
of  fine  salt,  and  10  bushels  of  fine  salt  as  much  as  7  bushels  of  coarse, 
and  7  bushels  of  coarse  salt  as  much  as  4  bushels  of  sand,  how  many 
bushels  of  sand  will  weigh  as  much  as  40  bushels  of  wheat  ? 
15X7    X4X40s  ushek     AM 

14  X  10  X  7  T 

10* 


114  PERCENTAGE, 


PERCENTAGE. 


Pure  percentage,  or  percentage,  is  a  rate  by  the  hundred  of  a 
part  of  a  quantity  or  number  denominated  the  principal,  or  basis. 
But  percentage,  considered  as  a  means,  and  as  commonly  applied, 
is  mixed  and  related  in  an  eminent  degree ;  and  in  this  light  may 
be  regarded  as  divided  into  orders  bearing  different  names. 

Thus  Interest  is  percentage  related  to  intervals  of  time  in  the 
past. 

Discount  is  percentage  related  to  interest,  and  intervals  of  time 
in  the  future. 

Profit  and  Loss  is  comparative  percentage,  or  percentage  related 
to  the  positive  and  negative  interests  in  business,  etc.,  etc. 

Pure  percentage  is  commonly  called  brokerage  when  paid  to 
a  broker  for  services  in  his  line. 

It  is  called  commission  when  paid  to  or  received  by  a  factor 
or  commission  merchant  for  buying  or  selling  goods. 

It  is  called  premium  by  an  insurance  company,  when  taken  for 
insuring  against  loss. 

It  is  called  primage  when  it  is  a  charge  in  addition  to  the 
freight  of  a  vessel,  etc. 

Comparative  percentage  relates  to  the  differences  of  quantities, 
and  is  confined  always  to  the  idea  of  more  or  less.  It  implies  ratio. 
This  description  of  percentage,  though  much  in  practice,  seems  not 
to  be  well  understood ;  and  often  a  quantity  is  indirectly  stated  to 
be  many  times  less  than  nothing,  or  many  times  greater  than  it  is. 
The  difference  of  two  quantities  cannot  be  as  great  as  a  hundred 
per  cent,  of  the  greater,  however  widely  unequal  the  quantities 
may  be,  nor  as  small  as  no  per  cent,  of  the  greater  or  lesser,  how- 
ever nearly  equal  they  may  be.  No  quantity  or  number  can  be  as 
small  as  1  time  less  than  another  quantity  or  number ;  and  there- 
fore cannot  be  as  small  as  100  per  cent.  less.  But,  since  one  quan- 
tity may  be  many  by  1  time,  or  many  times  greater  than  another 
with  which  it  is  compared,  it  may  be  said  to  be  many  by  100  times, 
or  many  hundred  per  cent,  greater. 

When  one  of  two  quantities  in  comparison  is  stated  to  be  three 
times  less,  or  three  hnndred  per  cent,  less,  for  instance,  than  the 
other,  the  expression  is  incorrect  and  absurd.  The  meaning  evi- 
dently is,  that  it  is  two-thirds  less,  or  only  one-third  as  large  as  the 
other,  —  that  it  is  66$  per  cent  less,  or  only  33£  per  cent,  as  large 
as  the  other.  In  common  comparison,  1  is  the  measuring  unit.  In 
percentage,  100  is  the  measuring  unit. 


PERCENTAGE.  115 

Let  a  =  principal. 
b  =  percentage. 

s  =  amount  (sum  of  the  principal  and  percentage). 
d  =  difference  of  the  principal  and  percentage. 
r  =  rate  of  the  percentage. 
p  =  rate  per  cent,  of  the  percentage. 

a  =  s— -  b  =  b -r- r=z  100b  ^-pz=z  100s -^-  (100  -j-jp), 

6  =s —  a—arzzzap-±- 100, 

p  =  1  OOr  =1006  -f-  a  =  100(s  —  a)  -^  a, 

r  =;?  -f-  100  =  6  -J-  a  =  (.*  —  a)  -7-  a, 

s  =  a  -|-  6  =  a(l  +  r)  =  a(\ 00  -+-  p)  ~  100, 

d  =  o  —  6  =  2a  —  5  :=  s  —  2b  =  a(l  —  r). 

To  find  the  Percentage. 

EXAMPLES. 

What  is  I  of  1  per  cent,  of  $200  ? 

6  =ar  =  op -^-100  =  $0.50.     ^ras. 
^  of  2  per  cent,  of  50  is  what  part  of  50  ? 
50X8X2 
7X100     =**•     ^ 
What  is  I  of  I  of  \  of  24  per  cent,  of  150  lbs.  ? 
150  X  12  ~  100  =  18  lbs.     Ans. 
What  is  2|  percent,  of  19  bushels  ? 

■¥  X  ^  =  0.45125  bushels.     ^Ins. 

Bought  a  job  lot  of  merchandise  for  $850,  and  sold  it  the  same 
day,  brokerage,  2\  per  cent.,  for  $975 ;  what  was  the  net  gain? 
s  —  sr —  a  =  s  —  (sr -f-  a)  =  s(l  — r)  — a  =  975  —  975  X  -025 

—  850  =  $100,625.     Jns. 

To  find  the  Rate  or  Bate  Per  Cent. 

EXAMPLES. 

What  per  cent,  of  $20  is  $2  ? 

r  =  b-^-a,p=zl00b—-az=.  10*  per  cent.    Ans. 
12  dozen  is  equal  to  what  per  cent,  of  2  dozen  ? 
12  —  2  =  6,  600  per  cent.    Ans. 


116  PERCENTAGE. 

What  part  of  5£  lbs.  is  f  of  2  lbs.  ? 

r=J  X  A  =  «=0.27A-     iin«. 

24£  per  cent,  is  what  per  cent,  of  36f  per  cent.  ? 

66 f  per  cent.     Ans. 

For  an  article  that  cost  $4,  $5  were  received ;  what  per  cent, 
of  $4  was  received  ? 

p  =  5  X  100  -J- 4  =  125  per  cent.     Ans. 

A  farmer  sowed  4  bushels  of  wheat,  which  produced  48  bushels ; 
what  per  cent,  was  the  increase  ?  48  is  more  than  4  by  what  per 
cent,  of  4  ?     The  difference  of  48  and  4  is  what  per  cent,  of  4  r 

a  —  b       a  100(a  —  b)       48  —  4       AQ   .    .       . 

100(48  —  4)  -f-  4  =  1100  percent.     .4  ns. 

What  per  cent,  would  have  been  the  decrease,  if  he  had  sowed 
48  bushels,  and  harvested  only  4  bushels  ?  4  is  less  than  48  by 
what  rate  of  48  V  The  difference  of  48  and  4  is  what  per  cent,  of 
48? 

r=z(a  —  b)-r-a  =  l =  0.91§,  or  91§  per  cent.     Ans. 

Since  water  is  composed  of  8  atoms  of  oxygen  and  1  atom  of 
hydrogen,  what  per  cent,  of  it  is  oxygen  ?  8  is  what  per  cent, 
of  the  sum  of  8  and  1  ? 


=  1-       *     ,p  =  -^  =  At  =  -8889-, 


'—  a+&~~  a-f&'^a-j-ft- 8  +  1 

or  88.89  -  per  cent.     Ans. 

What  per  cent,  of  it  is  hydrogen  ?  1  is  what  pe^r  cent,  of  the 
sum  of  8  and  1  ? 

a  b  100b  1 

a-\-b      a-\-b    r       a-\-b      8  -f- 1 

11.11  -f-  per  cent.     Ans. 

How  many  volumes  of  water  must  be  added  to  100  volumes  of 
90  per  cent,  alcohol  to  reduce  it  to  50  per  cent,  alcohol  or  common 
proof?  90  is  more  than  50  by  what  per  cent,  of  50  ?  The  differ- 
ence of  90  and  50  is  what  per  cent,  of  50  ? 

(a  — 6)100       (90  —  50)100       OA        . 
P=  b  =  50  =8°'     An8' 


PERCENTAGE.  117 

How  many  volumes  of  50  per  cent,  akohol  must  be  added  to 
100  volumes  of  90  per  cent,  alcohol  to  produce  80  per  cent,  alcohol  ? 
90  is  more  than  80  by  what  per  cent,  of  the  difference  of  80  and 
50  ?  The  difference  of  90  and  80  is  what  per  cent,  of  the  differ- 
ence of  80  and  50  ? 

(a-i)100  =  (90-80)100  =  3 
*  b  —  V  80— -50  * 

How  many  volumes  of  90  per  cent,  alcohol  must  be  added  to  100 
volumes  of  50  per  cent,  alcohol  to  raise  it  to  80  per  cent,  alcohol  ? 
50  is  less  than  80  by  what  per  cent,  of  the  difference  of  90  and  80  ? 
The  difference  of  80  and  50  is  what  per  cent,  of  the  difference  of 
90  and  80  ? 

0-y)ioo= go^o)ioo  =  300,   Am. 

a  —  o  90  —  80 

If  to  2  volumes  of  95  per  cent,  alcohol,  1  volume  of  50  per  cent, 
alcohol  be  added,  what  per  cent,  alcohol  will  be  the  mixture  ?  The 
sum  of  50  and  twice  95  is  what  per  cent,  of  the  sum  of  2  and  1  ? 

2a  +  b       2X95  +  50       D/% 

y^y  = 2  +  l       =80  per  cent.    Ans. 

In  a  barrel  of  apples,  the  number  of  sound  ones  was  60  per 
cent,  greater  than  the  number  that  were  damaged.  What  per 
cent,  less  was  the  number  that  were  damaged  than  the  number 
that  were  sound  ?  60  per  cent,  is  what  per  cent,  of  the  sum  of 
100  per  cent,  and  60  per  cent.  ?     .6  is  what  rate  of  1  +  .6  ? 

a  .         100  O.a        m         1  60 

=  .375,  or 


l-{-a  1-}- a      l+.a  l.a      1  +  60 

37^  per  cent.    Ans. 

Since  the  number  of  damaged  apples  was  37£  per  cent,  less  than 
the  number  that  were  sound,  what  per  cent,  greater  was  the  num- 
ber that  were  sound  than  the  number  that  were  damaged  ? 

r  =  a  +  (1  —  a)  =  1  +  (1  —  a)  —  1  =  60  per  cent.    Ans. 

Since  the  number  of  sound  ones  was  60  per  cent,  greater  than 

the  number  that  were  damaged,  what  per  cent,  of  the  whole  were 

sound  ? 

a  +  a*       l  +  .a            100  +  60       OA  . 

r  =  — -L—  =— L_  ,  pr=z J =80  per  cent.    Ans. 

What  per  cent,  of  the  whole  were  damaged  ? 

(100  —  60)  +  2  =  20  per  cent.     Ans. 


118  PERCENTAGE. 

Since  20  per  cent,  of  fhe  apples  were  damaged,  what  per  cent, 
less  was  the  number  that  were  damaged  than  the  number  that  were 
sound  ? 

1  —  2. a                    1                 100  — 2a      _„            100 
r=z =  1 ,  v= =  100 -  = 

2  — 2. a  2  —  2. a  9F      200  — 2a  200  —  40 

37^  per  cent.     Ans. 

What  per  cent,  greater  was  the  number  that  were  sound  than 
the  number  that  were  damaged  ? 

r  =  2  —  (1  -}-2.a)  =  2  —  2. a  —  1  =  60  percent.     Ans. 

Since  80  per  cent,  of  the  whole  were  sound,  what  per  cent,  less 
was  the  number  that  were  damaged  than  the  number  that  were 
i  sound? 

2. a—  1       ,         1         2X-80  —  1        M1  . 

=  1 =  -^ =  37£  per  cent.    Ans. 


2. a  2. a       2X-80 

Since  the  number  of  damaged  ones  was  3  7£  per  cent,  less  than 
the  number  that  were  sound,  what  per  cent,  of  the  whole  were 
sound? 

1  100  100 

80  per  cent.    *Ans. 


2  —  2.air      2  — 2a       2  —  2X37.5 

Since  80  per  cent,  of  the  whole  were  sound,  what  per  cent 
greater  was  the  number  that  were  sound  than  the  number  that 
were  damaged  ? 

r  sa     ~",a  =  2.a  —  1  =  2  X  .80  —  1  =  60  per  cent.     Ans. 
2 

Lost  20  per  cent,  of  a  cargo  of  coal  by  jettison,  and  5  per  cent, 
of  the  remainder  by  screening,  what  per  cent,  of  the  coal  was 
saved  ? 

a  —  b'=d'  I  r  =  (l  —  r')  (1  —  r")  =  (l—  .20)—  (1  —  .20) 
d'—b"  =  d"$  X  .05  =(1—  .20)(1  —  .05)  =  76  per  cent.  Ans. 
d"  —  b'"  =  d"',8ic. 

Yesterday  drew  12  per  cent,  of  my  balance  of  $1,2 73  in  the 
bank,  and  deposited  $1,000;  and  to-day  have  drawn  31|  per  cent, 
of  the  balance  left  over,  or  as  it  stood  last  night.  What  per  cent  of 
the  sum  of  the  first-mentioned  balance  and  deposit  of  yesterday 
have  I  drawn  ? 

b'-\-b"       512  +  1487.575        Q7  OQr,    .  .       A 

r  =z  — L = • — . =  37.9354  4-  per  cent    Ans. 

a-\-m  4273  +  1000  '  * 


PERCENTAGE.  119 

Wli.it  per  cent,  of  the  said  sum  is  remaining  in  the  bank  ? 

fr'.|-6"_a-J- m  —  V  —  b" _u  +  m—  (7/ -f  h") 
7+£~         a4-i»  ~^fm  =62.0646- 

1  '  l  per  cent.     Ans. 

What  per  cent,  predicating  it  upon  the  first-mentioned  balance, 
have  I  drawn  ? 

b'4-b"       512.76 -f- 1487.576        ,CQ104  .        . 

r=  — X —  ss !— — =  46.81 34  -  per  cent.     Ans. 

a  4273  ■ 

What  per  cent,  have  I  drawn,  predicating  it  upon  what  I  now 
have  in  the  bank  ? 

b'+w h!_ztl"       — 

r— ^-ZF+^=-T''-~  o+»—  £&l+&")~~  61.1225  -f 

percent.     vlns. 
What  amount  of  money  must  I  deposit  to  makfe  good  62^  per 
cent,  of  the  aforementioned  sum  V 

d=zr  (a  -\-  m)  +V  -\-b"  —  (a  +  m)  =r  (a  +  w)—  d"  = 

$22.96.     Ans. 

To  Jind  the  Principal  or  Basis. 

EXAMPLES. 

The    percentage  being   250,   and    the  rate  .06,  what   is  the 
principal  ? 
a  =  b^-r=  100b -^-p  =  250-^.06  =  25,000-^-6  =4,1 66f.    Ans. 

A  tax  at  the  rate  of  £  of  1  per  cent,  on  the  valuation  was  $27.50. 
What  was  the  valuation  ? 

n  =  tX6X100=>  Anfi 

5 

Sold  120  barrels  of  flour,  which  amounted  to  12  per  cent,  of  a 
certain  consignment.  The  consignment  consisted  of  how  many 
barrels  ? 

120-^0.12  =  1,000.     An*. 

216  bushels  is  more  by  8  per  cent., or  8  percent,  more,  than  what 
number  of  bushels  ?  8  per  cent,  more  than  what  number  is  equal 
to  216  ?     What  number,  plus  8  per  cent,  of  it,  will  make  216  ? 

a  =  s-r(l~L-r)  =  216-7-1.08  =  200.     Ans. 

200  lbs.  is  less  by  8  per  cent.,  or  8  per  cent,  less,  than  what  num- 


120  INTEREST. 

ber  of  lbs.  ?    8  per  cent,  less  than  what  number  is  200  ?    What 
number,  minus  8  per  cent,  of  it,  is  equal  to  200  ? 

a  =  d  -±-  (1  —  r)  =  200  ~  (1—  .08)  =  21 7fa     Ans. 

.-.  217^—  217^X.08  =  200=a  —  b  =  d=a(l—  r). 

To  a  quantity  of  silver,  a  quantity  of  copper  equal  to  20  per 
cent,  of  the  silver  is  to  be  added,  and  the  mass  is  to  weigh  22 
ounces.     What  weight  of  silver  is  required  ? 

a  =  5-7-  (l-}-r)  =  22-^-1.2  =  18£  ounces.     Ans. 

What  weight  of  copper  is  required  ? 

s  —  — . —  =  r-7—  =  34  ounces.    Ans. 
1  -f-  r       1-f-r        8 

To  a  quantity  of  copper,  a  quantity  of  nickel  equal  to  62£  per 
cent,  of  the  copper,  a  quantity  of  zinc  equal  to  33£  per  cent,  of  the 
copper,  and  a  quantity  of  lead  equal  to  5  per  cent,  of  the  copper, 
are  to  be  added;  and  the  whole  is  to  weigh  40£  pounds.  The 
weight  of  each  constituent  of  the  alloy  is  required. 

_  s  _ 40$ 

a—  i  j^r  +  ri  +  rH  ~  i  _|_.  62^+.  33£+.  05 

=r  20  lbs.  of  copper, 
b  =  20  r  =  1 2£  lbs.  of  nickel,    . 
6'=20r'=6§lbs.ofzinc, 
br,  _  20  r"  =  l  lb.  of  lead. 


INTEREST. 

Universal  for  any  rate  per  cent. 

T  =  time  in  months  and  decimal  parts  of  a  month ;  t=  time  in  days ; 
P  as  principal  ;  r  =  rate  per  cent.,  expressed  decimally;  i= interest. 

PXTXr      TxtXr 


12  365 


p_i2_t_365i  T_i2_*  j_365*        xti^aegj 

—  Tr~~"   tr  '       ~~Pr*       ~Pr'       "~"PT       Ft  ' 
Example.  —  A  promissory  note,  made  April  27,  1864,  for 


INTEREST.  121 

$825 A^j-  and  interest  at  6  per  cent,  matured  Oct.  6,  1865  :  what 
was  the  interest? 


Oct.  is  10th  month. 
April  is  itii  month. 

j:         m,        d. 

1865  .  10  .     6 

'64  .     4  .  27 


Time  from  April  27  to  Oct.  6  (one  of 
the  dates  always  included)  ss  162  days, 
which,  added  to  the  865  days  in  the  year 
preceding  =  527  days. 

Note.  — One  day's  interest  at  least  is  gener- 
ally lost  by  computing  the  time  in  years  and 
months,  or  months,  instead  of  days. 


Time=       1.5.9 

825.25  X  17.3  X -06 —  12  =  $71.38.     Ans. 
825.25  X  527  X-06  -7-  365  =  $71.49.     Ans. 

To  find  a  constant  divisor,  fc,  for  any  given  rate  -per  cent* 

When  the  time  is  taken  in  months,  k  =  1 2  -7-  r. 

When  the  time  is  taken  in  days,  k  s=  365  -7-  r  ;  thus, 

P  X  t 
When  the  rate  is  6  per  cent.  -^-^-= Interest. 

P  X  t 

When  the  rate  is  7  per  cent.       ~     sa  Interest,  &c. 

Example.  —  Required  the  interest  on  $750  for  93  days,  at  7 
per  cent. 

750  X  93  -7-  5214  =  $13.38.     Ans. 

Example.  —  What  is  the  rate  per  cent,  when  $450  gains  $94} 
in  3  years  ? 

450  :  100  *.:  94.5  :  3x=7  per  cent.     Ans. 

94.5  -f-  3  X  450  =  .07.     Ans. 

Example.  —  In  what  time  will  $125  at  6  per  cent,  gain  $18|? 

6  :  100  ;:  18.75  :  125  X  »=  2}  years.     Ans. 

18.75  -f-  125  X-06  =  2}  years.     Ans. 

Example.  —  What  principal  at  5  per  cent,  interest  will  gain 
$16}  in  18  months? 

5  :  100  ;:  16.875:  1.5  X  a:  =  8225.     Ans. 

16.875  X  12-f-18  X-05  =  $225.     Ans. 
11 


122  COMPOUND   INTEREST. 

WJien  partial  payments  have  been  made. 

Rule.  —  Find  the  amount  (sum  of  the  principal  and  interest) 
up  to  the  time  of  the  first  payment,  and  deduct  the  payment  there- 
from ;  then  find  the  interest  on  the  remainder  up  to  the  next  pay- 
ment, add  it  to  the  remainder,  or  new  principal,  and  from  the  sum 
subtract  the  next  payment ;  and  so  on  for  all  the  payments ;  then 
find  the  amount  up  to  the  time  of  final  payment  for  the  final 
amount. 


COMPOUND  INTEREST. 

If  we  calculate  the  interest  on  a  debt  for  one  year,  and  then  on 
the  same  debt  for  another  year,  and  again  on  the  same  debt  for 
still  another  year,  the  sum  will  be  the  simple  interest  on  the  debt 
for  three  years.  But,  on  the  contrary,  if  we  calculate  the  interest 
on  the  debt  for  one  year,  and  then  on  the  amount  (sum  of  the  prin- 
cipal and  interest)  for  the  next  year,  and  then  on  the  second 
amount  for  the  third  year,  the  sum  of  the  interest  so  calculated 
will  be  the  compound  interest,  or  yearly  compound  interest,  on  the 
debt  for  three  years ;  equal  to  the  simple  interest  on  the  debt  for 
three  years,  plus  the  yearly  compound  interest  on  the  first  year's 
interest  for  two  years,  plus  the  simple  interest  on  the  second  year's 
interest  for  one  year.  So,  if  we  divide  the  time  into  shorter 
•periods  than  a  year,  and  proceed  for  the  interest  as  last  suggested, 
the  interest  will  be  compound.  Thus  we  have  half-yearly  com- 
pound interest,  or  compound  interest  semi-annually,  quarter- 
yearly  compound  interest,  or  compound  interest  quarterly,  &c. 

This  method  of  computing  interest  is  predicated  upon  the 
natural  idea,  that  interest,  when  it  becomes  due  by  stipulation  and 
is  withheld,  commences  to  draw  interest,  and  continues  at  use  to 
the  holder,  at  the  same  rate  as  the  principal,  until  it  is  paid,  like 
other  over-due  demands;  and  that  the  interest  so  made  matures 
and  becomei  due  as  often,  and  at  the  same  periods,  as  that  on  the 
principal. 

It  will  be   perceived   by  the   foregoing   that   the   >rorkin(/-t>'me   in 

compound  interest  is  the  interval  between  th«*  stipulated  payments 

of  toe  interest,  or  between   one  stipulated  payment  of  the    interest 

and  that  of  another;  and  that  the  wurkiiKj-rate  is  pro  rata  to -the 
rate  per  annum. 

Thus  the  amount  of  $10o  at  semi-annual  compound  interest  for 
2  years,  at  G  per  cent,  per  annum,  is 


COMPOUND    INTEREST.  123 

100  X  (1.03)*  =  $112.550881  =$112.55,  or 
100. 
.03 

3. 

100. 

103. 
.03 

3.09 
1031_ 

106^09 
.03 


3.1827 
106.09 

109.2727 
.03 

3.278181 
109.2727 

SI  12.550881,  as  before. 

If  we  let  P  =  principal  or  debt  at  interest, 
r  =  working-rate  of  interest, 

n  =  number  of  intervals  into  which  the  whole  time  is 
divided  for  the  payment  of  interest,  or  number  of  consecutive 
intervals  for  the  payment  of  interest  that  have  transpired  without 
a  payment  having  been  made, 

i  =  compound  interest, 
A  =  P  -j-  i  or  amount,  then 

A  =  P(H-r).;P=^;r=^-li 

^-=(l  +  r)»;«  =  A-P. 

Example.  —  What  is  the  compound  interest,  or  yearly  com- 
pound interest,  on  $100  for  1£  years,  at  6  per  cent,  a  year  ? 

100X  1.06X  1-03  =  109.18  —  100  =  $9.18.     Ans. 

Example. — What  is  the  amount  of  $560.46,  at  7  percent, 
compound  interest  per  year,  for  6  years  and  57  days  ? 

560.46  X  (1.07)6  x(l  +  '°73^557)  =  $850.29.     Ans. 


124  COMPOUND   INTEREST. 

Example.  —  The  principal  is  $250,  the  rate  8  per  cent,  a  year, 
the  time  2  years,  and  the  interest  compound  per  quarter  year: 
required  the  amount. 

250  X  (l.  —  )    =$292.91.    Ans. 


('•t)8= 


When  Partial  Payments  have  been  made. 

Rule.  —  Find  the  amount  up  to  the  first  payment,  and  deduct 
the  payment  therefrom  ;  then  find  the  amount  up  to  the  next  pay- 
ment, and  therefrom  deduct  that  payment ;  and  so  on  for  all  the 
payments ;  then  find  the  amount  up  to  the  time  of  final  payment, 
tor  the  final  amount. 

Example. —  A  note  of  hand  for  $500  and  interest  from  date, 
at  6  per  cent,  a  year,  has  been  paid  in  part  as  follows ;  viz.,  two 
years  and  four  months  from  the  date  of  the  note,  by  an  indorse- 
ment of  $50 ;  and  three  years  from  that  indorsement,  by  an  in- 
dorsement of  $150.  It  is  now  eight  months  since  the  last  payment 
was  made,  and  the  demand  is  to  be  settled  in  full :  required  the 
amount  at  the  present  time,  interest  being  compound  per  year. 

500  X  (1.06)2  x  1.02  —  50=523.036 

(1.06)8 

622.944 
150 


472.944 
1.04 


$491.86.     Ans. 

The  following  table  shows  (1  -f-  r)  raised  to  all  the  integer 
powers  from  1  to  12  inclusive  ;  r  being  taken  at  4,  5,  6,  7,  8,  and  10 
per  cent.  If  the  numbers  in  the  column  headed  years  are  taken 
to  represent  years,  then  4  per  cent.,  5  per  cent.,  &c,  at  the  head 
of  the  columns  of  powers,  will  stand  for  per  cent,  per  annum :  if 
they  are  taken  to  represent  half-years,  then  4  percent,  5  percent., 
&c,  will  stand  for  per  cent,  per  half-year,  &c.  The  quantities  in 
the  columns  are  powers  of  (1  -f~r)>  °f  ^hich  the  numbers  referred 
to  and  standing  opposite,  respectively,  are  the  exponents.  Thus, 
1.26248,  in  the  6  per  cent,  column,  and  against  4  in  the  column 
marked  vears,  =  (1.06)4  ;  and  so  with  the  others.  The  powers 
or  quantities  in  the  columns  are  co-efficients  in  the  calculations. 


COMPOUND    INT! 


125 


Years. 

4  p«r  cent. 

5  per  cent. 

e  per  cent. 

7  per  cent. 

S  ]..T  C.llt. 

io  percent 

1 

1.04 

1.08 

1.06 

1.07 

1.08 

1.10 

2 

1.081G 

1.1025 

l.ri:;.; 

1.1449 

1.1664 

1.21 

3 

1.12486 

1.15762 

1.11)102 

1.22504 

1.25971 

1.881 

4 

1.16986 

1.21551 

1.26248 

1.3108 

1.36049 

1.4641 

5 

1.21668 

1.27628 

1.38828 

1.40255 

1.46933 

1.61051 

6 

1.26532 

1.3401 

1.41852 

1.50073 

1.58687 

1.77156 

7 

1.31593 

1.1071 

1.50363 

L.60578 

1.71382 

1.94872 

8 

1.36857 

1.47746 

1.59385 

1.71819 

1.85098 

2.14359 

9 

1.42331 

1.55133 

1.68948 

1.88846 

1.999 

2.35795 

10 

1.48024 

1.62889 

1.79085 

1.96715 

2.15892 

2.59374 

11 

1.53945 

1.71034 

1.8983' 

2.10485 

2.33164 

2.85812 

12 

1.60103 

1.79586 

2.0122 

2.25219 

2.51817 

3.13843 

Note.  —  If  a  co-efficient  is  wanted  for  a  greater  number  of  years  or  intervals 
of  time  than  is  given  in  the  table,  square  the  tabular  co-efficient  opposite  half 
that  number  of  intervals,  or  cube  the  tabular  co-efficient  opposite  oue-tbird 
that  number  of  intervals,  &c,  for  the  co-efficient  required.    Thus, 

1. 91)92=1. 586S73=1. OS  12X  1.08°=  1.08^  =  3.990, 

the  co-efficient  for  18  years  or  intervals  at  8  per  cent,  per  interval,  &c. 

It*  the  compound  interest  alone  is  sought  on  a  given  principal,  subtract  1 
from  the  tabular  power  corresponding  to  the  time  and  rate,  and  multiply  the 
remainder  by  the  given  principal  ;  the  product  will  be  the  compound  interest. 
Thus  (1.26532— 1)X  100  =  $26,632,  the  yeprly  compound  interest,  at  4  per 
cent,  per  annum,  on  $100  for  0  years,  or  the  half-yearly  compound  interest,  at 
8  per  cent,  per  annum,  on  $100'i'or  :5  years,  or  the  half-yearly  compound  inter- 
est, at  4  per  cent,  per  half  year,  on  $100  for  0  half-years. 

Example.  —  What  is  the  amount  of  $125.54,  at  5  per  cent, 
compound  interest,  for  7  years,  21  days? 

21  X   05 

1-1 ~; — =z  1.00288,  the   co-efficient  for  the   odd  days;  and, 

1        365 

turning  to  the  5  per  cent,  column  in  the  table,  we  find  against  7,  in 

the  column  of  years,  1.40  71,  the  co-efficient  for  7  years  :  then 

1 25.54  X  1.4071  X  1.00288  =:  $1 78.20.     Ans. 

Example.  —  In  -what  time,  at  7  per  cent,  compound  interest 
per  annum,  will  $1000  gain  $462?  A-^-P=r  (1  -f-r)n  :  then 
146 2  -j- 1000  =  1.462,  the  co-efficient  demanded.  Turning  now 
to  the  7  per  cent,  column  in  the  table,  we  find  the  nearest  less 
co-efficient  there  (there  being  none  that  exactly  corresponds)  to 

be  that  for  5  years  ;  viz.,  1.40255.     And  (     4Q255  —  !)  "T-  -07  = 
.60553,  the  fraction  of  a  year  over  5  years  to  the  answer. 

.60553  X  365  =  221  days:  5  years,  221  days.     Ana. 
11* 


126 


COMPOUND   INTEREST. 


The  following  table  is  of  the  same  nature  as  the  preceding, 
and  is  applicable  when  the  interest  becomes  due  at  regular  inter- 
vals short  of  a  year,  or  when  the  working-rate  in  compound  inter- 
est is  less  than  4  per  cent. 

The  quantities  in  the  If  per  cent,  column  apply  to  quarter-yearly 
compound  interest  when  the  rate  is  7  per  cent,  a  year ;  and  those 
in  the  1£  per  cent,  column,  to  quarterly  compound  interest  when 
the  rate  is  5  per  cent,  a  year ;  also  the  former  are  applicable  to 
monthly  compound  interest  at  21  per  cent,  per  annum,  and  the 
latter  to  monthly  compound  interest  at  15  per  cent,  per  annum ; 
and  so  relatively,  throughout  the  table. 


1 

u 

a 

s 

1 

■ 

s 
8 

•J 

a 
S 

■ 

i 

1 

1 

% 

V 

I 

B 

1 

1 

1 

1 

s 

A 

S 

ft 

M 

£ 

£ 

3 

ft 

* 

1 

1.035 

1.03 

1.025      1.02 

1.0175 

1.015 

1.0125 

1.01 

1.005 

2 

1.07123  1.0609 

1.05063  1.0404 

1.03531  1.03023 

1.025161.0201 

1.01003 

3 

1.10872  1.09273 

1.076891.06121 

1.05342- 1.04568 

l.o:37L»7  1.0303 

1.01508 

4 

1.1475211.12561 

1.10381 1.08243 

1.07186il.0t3136 

1.05095  1.0406 

1.02015 

6 

1.1876911.15927 

1.13141 1.10408 

1.09062 

1.07728 

1.06408 

1.05101 

L.02525 

6 

1.22925 

1.19405 

1.15969.1.12616 

1.1077 

1.09344 

1.0774 

1.06152 

1.03038 

7 

1.27228 

1.22987 

1.18869|l.l4869 

1.12709 

1.109S4 

1.09087 

1.07214 

8 

1.31681 

1.26677 

1.2184  11.17166 

1.14681 

1.12G49 

1.10451 

1.08888 

1.04071 

9 

1.3629 

1.30477 

1.248861.19509 

1.16688 

1.14339 

1.11831 

1.09369 

1.04591 

10 

1.4106 

1.34392 

1.280081.21899 

1.1873 

1.16054 

1.18229 

L10462 

1.05114 

11 

1.45997 

1.38423 

1.312091.24337 

1.20808 

1.17795 

1.146461.11567 

1.0564 

12    1.51107  1.42576 

1.34489!l.26824 

1.22922 

1.19562 

1.160781.12683 

1.06168 

Example.  —  What  is  the  amount  of  $750  for  4  years  and  40 
days,  allowing  half-yearly  compound  interest,  at  7  per  cent,  a  year  ? 

In  this  case,  the  working-rate  for  the  full  periods  of  time  is  3£  per 
cent,  and  there  are  8  such  full  periods  ;  then,  seeking  the  co-efficient 
in  the  3£  per  cent,  column,  we  find  against  8,  in  the  column  of 

times,  the  quantity  or  co-efficient  1.31681 ;  and  1  -f-  — ~ —  = 

1.00767  :  therefore 

750  X  1.31681  X  1.00767  =  $995.18.     4ft* 

Example.  —  What  is  the  amount  of  $1000  at  compound  inter- 
est per  quarter-year,  at  1£  per  cent,  per  quarter-year,  for  4£  years  ? 

1000  X  1.126492  X  1.015  =  $1288.01.     Ans. 


HANK    INTEltKST.  127 


BANK  INTEREST  OR  BANK  DISCOUNT. 

A  bank  loans  money  on  a  promissory  note  made  payable  with- 
out interest  at  a  future  period.  The  operation  is  called  discounting 
the  note  at  bank,  and  is  as  follows :  The  bank  takes  the  note,  funis 
the  interest  on  it  for  three  days  more  time  than  by  its  own  tenor  it 
has  to  run,  subtracts  it  from  the  principal,  and  hands  the  balance, 
called  the  avails  of  the  note,  in  its  own  bills,  to  the  party  soliciting 
the  loan,  or  offering  the  note  for  discount,  as  it  is  called  ;  whereby 
the  note  becomes  the  property  of  the  bank,  and  the  maker  and 
indorscrs  are  held  for  its  payment  when  it  matures. 

The  three  days  mentioned  are  called  days  of  grace,  and  the 
note  does  not  become  due  to  the  bank  until  three  days  after  it 
becomes  due  by  its  own  tenor.  These  proceedings  are  sanctioned 
by  usage,  and  protected  by  law. 

Bank  interest,  then,  is  bank  discount,  and  bank  discount  is  bank 
interest.  But  bank  discount  is  not  discount,  nor  is  it  what  is  called 
legal  interest  on  the  money  loaned.  It  is  the  interest  on  the  money 
loaned,  plus  the  interest  on  the  interest  of  the  loan,  plus  the  inter- 
est on  the  difference  of  the  sum  taken  and  the  interest  on  the  loan 
for  the  time  of  the  loan  !  A  kind  of  interest  more  onerous,  if  any 
description  of  interest  be  onerous,  than  compound  interest,  rate 
for  rate  and  time  for  time,  as  may  be  readily  perceived. 

Let  P  =  principal  or  face  of  the  note. 

r  =  working-rate  of  the  interest  for  the  time  of  the  loan. 
a  =  avails  of  the  note  or  sum  borrowed. 
i  =  bank  interest. 
t  =  time  of  the  loan. 

R :  r : :  T :  t.  R  being  the  rate  per  cent,  per  annum,  and  T 
one  year. 

P  =  a-Hl—  r).     a  =  Y  —  Pr.     i  =  Vr.    r  =  (P  —  a)-j-P. 

If  we  let  n  represent  the  time  of  the  note  in  months, 
_  R  n  ,   3  R 

r —  ~Y2  ~T  ggj='     But  it  is  the  practice  with  many  banks  to  count 

the  days  of  grace  as  so  many  3G0ths  of  a  year. 

Putting  d  to  represent  the  time  of  the  note  in  days, 

Rrf-f3R  .         . 

r  = ,  true  time  and  rate. 

365 

"With  some  banks,  it  is  the  practice,  in  calculating  interest,  to  take 
the  time,  when  it  does  not  exceed  93  days,  as  so  many  360ths  of  a 
year. 

A  note  having  3  months  to  run  from  Aug.  10,  for  instance,  will 


128 


BANK    INTEREST. 


fall  due  Nov.  10-13;  but  one  having  90  days  to  run  from  Aug. 
10  will  fall  Nov.  8-11.  The  time  including  grace  of  the  former 
is  3  mo.  3  ds.,  and  that  of  the  latter  3  mo.  2  ds.,  mean  time.  Never- 
theless, the  former  embraces  95  days,  or  one  day  more  than  mean 
time,  and  the  latter  but  93  days. 

The  following  table  shows  1  —  r,  mean  time,  for  the  intervals  of 
time  set  down  in  the  left-hand  column  ;  B,  being  taken  at  4,  5,  6,  7, 
and  8  per  cent,  per  annum,  as  set  down  at  the  top  of  the  columns. 


Time. 

4 

6 

6 

7 

8 

mo. 

da. 

per  cent. 

per  cent. 

per  cent. 

percent. 

per  cent. 

1 

3 

.996333 

.995417 

.9945 

.993583 

.992667 

2 

3 

.993 

.99125 

.9895 

.98775 

.986 

3 

3 

.989667 

.987083 

.9845 

.981917 

.979333 

4 

3 

.986333 

.982917 

.9795 

.976083 

.972667 

5 

3 

.983 

.97875 

.9745 

.97025 

.966 

6 

3 

.979667 

.974583 

.9695 

.964417 

.959333 

7 

3 

.976333 

.970417 

.9645 

.958583 

.952667 

8 

3 

.973 

.96625 

.9595 

.95275 

.946 

9 

3 

.969667 

.962083 

.9545 

.946917 

.939333 

10 

3 

.966333 

.957917 

.9495 

.941083 

.932667 

11 

3 

.963 

.95375 

.9445 

.93525 

.926 

12 

3 

.959667 

.949583 

.9395 

.929417 

.919333 

Putting  k  to  represent  the  tabular  quantity  1  —  r, 

a=  Pit,  P=^  a  -f-  fc,  i  =  P  —  a  =  P—  Pfc 

Example.  —  What  will  be  the  avails  of  a  note  for  $1,250 
payable  in  4  months  if  discounted  at  a  bank,  interest  being  7  per 
cent,  a  year  ? 

The  tabular  constant  1  —  r,  in  the  7  per  cent,  column,  against  4 
months  and  3  days  in  the  time  column,  is  .976083,  and 
$1,250  X  .976083  =  $1,220.10.     Ans. 

Example.  —  For  what  sum  must  I  make  a  note  having  6  months 
to  run,  in  order  that  the  avails  at  bank,  If  discounted  on  the  day 
of  the  date  of  the  note,  may  amount  to  $956.38,  interest  being  6 
percent,  per  annum? 

By  the  table,  $956.38  -f-  .9695  =  $986.4  7.     A  m. 

Example.  —  What  is  the  rate  of  bank  interest  when  the  nomi- 
nal or  legal  rate  is  7  per  cent.? 

.07  -f-  (1  —  .07)  =  .07527  =  7f  +  rffo  per  cent. 

.i    —  A  note    living  5  BMmthfl  1<>   run  from    Kelt.  1   will  fall  due  July 

tnd  Hi.   time,  lnelu<finc  graoe,  is  ■:,  mo.  8  da.      156  days,  mean  time. 

r.ui  the  time  In  days  from  Feb.  l  to  Julj  «.  when  February  has  but  iittdays, 

is  163  days  only,  or  2  days  short  of  mean  time. 


DISCOUNT.  —  COMI»OUND   DISCOUNT.  129 


DISCOUNT. 

Discount  is  a  deduction  of  the  interest  on  the  present  worth  or 
availability  of  a  debt  not  yet  due,  in  consideration  of  its  present 
payment  *  The  principal  is  the  present  nominal  value  of  the  debt, 
interest  included,  if  any  interest  lias  accrued  The  time  is  the 
interval  from  the  present  to  the  date  at  which  the  debt  will  become 
due.  The  rate,  is  the  legal  rate  of  interest,  if  no  other  rate  is  speci- 
fied; and  the  present  /corf//  is  that  sum  of  money,  which,  if  put  at 
interest  at  the  same  rate  and  for  the  same  time  as  the  discount,  will 
amount  to  the  principal. 

Let  a  represent  the  principal,  d  the  discount,  If  the  present 
worth,  and  i  the  interest  on  one  dollar  for  the  time  and  at  the  rate 
of  the  discount. 

w  =  a-±-  (1  -\-i)  =:  a  —  d.      d  =  fltf-f-  (1-|-  i)  —  a  —  w. 
a  =  d  (1  -j- 1)  -7-  i  =2  d  -j-  w. 

Example.  —  Required  the  discount  on  $250  for  8  months  at  6 
per  cent. 

The  interest  on  $1  for  8  months  at  6  per  cent,  is  .04  of  a  dollar, 
or  4  cts.  ^  and 

250  X -04-4- (1 -j-. 04)  =  $9.6154.     Ans. 

Example.  —  Required  the  present  worth  of  $1272.62  due  247 
days  hence,  discount  7  per  cent. 

The  interest  on  $1  for  247  days  at  7  per  cent.  =  247X-07  -f-365 
=  0.04737,  and 

1272.62-^- 1.04737  =$1215.06.     Ans. 

Note. — "  Talcing  off,  in  common  parlance,  a  certain  per  centum  from 
the  face  of  a  demand,  is  equal  to  deducting  the  interest,  at  that  rate  per  cent- 
turn,  on  the  present  worth  for  1  year,  plus  the  interest  on  the  interest  of  the 
present  worth,  at  the  same  rate  per  centum  for  1  year. 


COMPOUND  DISCOUNT. 

Compouxd  Discount  is  to  compound  interest  what  simple  dis- 
count is  to  simple  interest.  In  both  cases  of  discount,  the  differ- 
ence between  the  principal  and  the  discount  is  that  sum  of  money, 
which,  if  put  at  interest  for  the  same  length  of  time,  at  the  same 
rate,  and  in  the  same  general  manner  as  the  discouut,  will  amount 
to  the  principal. 

Rule.  —  Add  1  to  the  rate  per  cent,  of  the  discount  for  the 


130  COMPOUND   DISCOUNT. 

working-time,  and  raise  the  sum  to  a  power  corresponding  with  the 
number  of  working-times ;  divide  the  principal  by  the  power,  and 
the  quotient  will  be  the  present  worth ;  subtract  the  present  worth 
from  the  principal,  and  the  remainder  will  be  the  compound 
discount. 

Note. — The  tables  of  the  powers  of  1  +  r,  applicable  to  compound  in- 
terest, are  equally  applicable  to  compound  discount. 

Example.  —  Required  the  present  worth  of  a  debt  of  $250, 
allowing  yearly  compound  discount,  at  7  per  cent,  a  year,  for 
8  years  84  days. 

1  _l_  -7  X  8*  —  1.01611,  the  working-rate  for  the  84  days,  and 
'365  '  c  J  ' 

250  -f-  (1.078  x  1.01611)  b=  $200.84.     Ans. 

Example.  —  What  is  the  present  worth  of  a  debt  of  $150.25, 
due  3  years,  3  months,  and  10  days  hence,  without  interest,  allow- 
ing compound  discount  per  quarter-year,  at  1^  per  cent,  per  quar- 
ter-year ? 

150.25 -^(l.01513X  l.-06  X  10)=Ans. 
\  365       / 

By  table,     150.25-^(1.19562  X  1-015  X  1.00164)=      I 

$123.61.     Ans. 

Note.— What  is  here  denominated  the  debt,  or  principal,  represents  the 
debt  at  the  close  of  the  time  of  the  discount;  that  is,  if  the  debt  be  on  in- 
terest, the  interest  must  be  included  in  what  is  here  called  the  debt,  or 
principal. 


PROFIT  AND  LOSS. 

The  term  "Profit  and  Loss,"  as  intimated  in  treating  of 
Percentage,  relates  to  the  positive  and  negative  interests  in 
business,  and  embraces  the  idea  of  both. 

Both  profit  and  loss  are  absolute  quantities,  and  are  expressed  by 
the  difference  of  the  cost  price  and  selling  price  that  limit  them. 
They  are  usually,  however,  estimated  by  percentage,  predicated 
upon  the  first-mentioned  price  or  prime  cost. 

When  the  selling  price  is  greater  than  the  cost  price,  or  when 
tin-  money  obtained  by  the  disposal  of  property  exceeds  what  the 
property  cost,  the  difference  is  positive,  and  denotes  increase, 

profit,  or  gain.  Conversely,  when  the  cost  price  is  greater  than  the 
selling  price,  or  when  property  is  disposed  of  for  less  money  than 
it  cost,  the  difference  is  negative,  and  denotes  decrease,  loss,  or 


PROFIT   AND   LOSS.  131 

waste.  So,  the  difference  of  the  two  prices,  divided  by  the  cost 
price,  expresses  the  rate  of  gain  on  the  cost  when  the  selling  price 
is  the  greater, -—expresses  the  rate  of  loss  on  the  cost  when  tin- 
cost  price  is  the  greater. 

Let  c  represent  the  cost  price,  purchase  price,  par  value,  or  sum 
of  money  paid  for  the  property;  s,  the  selling  price,  trade  price, 
premium  price,  or  sum  of  money  received  in  exchange  for  the 
property ;  r,  the  rate  of  the  profit  or  loss ;  p,  the  rate  per  cent,  of 
the  profit  or  loss. 

To  find  the  rate  or  rate  per  cent,  of  the  profit  or  loss. 

r  ■=.  ■  ~    .    p  —  >.  ~   ' .    Moreover,  when  the  difference  is 

e  ■    *  c 

s                                                                  s 
positive,  r  = 1 ;  and,  when  it  is  negative,  r  —  1 

Example.  —  Paid  $4  for  an  article,  and  sold  it  for  $5.     What 

?er  cent,  was  gained  ?     5  is  more  than  4  by  what  per  cent,  of  4  ? 
ne  difference  of  5  and  4  is  what  percent,  of  4?    5  —  4  =  $1, 

gained;  and  —^—z=:. 25  =  £ — 1.     25  per  cent.     Ans. 

Example.  —  Paid  $5  for  an  article,  and  sold  it  for  $4.     What 

per  cent,  was  lost  ?     4  is  less  than  5  by  what  per  cent,  of  5  ?     The 

difference  of  4  and  5  is  what  per  cent,  of  5?     4  —  5=r  —  lz=$l, 

5  ~  4 
lost ;  and  — - —  =  .20  zzr  1  —  £.     20  per  cent.    Ans. 
o 

Example.  —  A  whistle  that  cost  3  cents  was  sold  for  20  cents  1 
The  profit  was  how  much  per  cent  ?  (20  ~  3)  -f-  3  =  5f  or  566§ 
per  cent.     Ans. 

Example.  —  A  fop  paid  $10  for  a  well-made  and  well-fitting 
pair  of  boots  for  his  own  wear,  that  were  worth  what  they  cost  him ; 
but,  being  told  that  they  were  unfashionably  large,  sold  them  for 
$4.  His  vanity  cost  him  what  per  cent,  of  the  purchase  price  ? 
1  —  ^=  .6  or  60  per  cent.     Ans. 

To  find  a  price  long  a  given  per  cent,  of  the  cost,  or  to  find  a  sell- 
ing price  that  shall  be  the  sum  of  the  cost  price  and  a  given  per 
cent,  of  it. 

s  =  c  -\-  cr  =  c  (1  -f-  r)  =  c  (100  -f-^)  -^-  100. 

Example.  —  At  what  price  must  I  sell  an  article  that  cost 
$2.35  to  gain  25  per  cent.  V  2.35,  more  25  per  cent,  of  it,  is  how 
much  ?  The  sum  of  $2.35  and  25  per  cent,  of  it  is  how  much  ? 
2.35  -4-  2.35  X  -25  =  2.35  X  1-25  =  $2.93f.     Ans. 


132  EQUATION   OF   PAYMENTS. 

To  find  a  price  short  a  given  per  cent,  of  the  cost,  or  to  find  a  sett- 
ing price  that  shall  be  the  difference  of  the  cost  price  and  a  given 
per  cent,  of  it. 

s=ic  —  cr=c(l  —  r)z=c  (100 — ;>)-^-100. 

Example.  —  I  have  a  damaged  article  of  merchandise  that  cost 
$2.75,  and  I  wish  to  mark  it  for  sale  at  30  per  cent,  below  cost. 
At  what  price  shall  I  mark  it  ?  2.75  less  30  per  cent,  of  it  is  how 
much?  The  difference  of  S2.75  and  30  per  cent,  of  it  is  how 
much?     2.75  (1—. 30)  =  2.75  X -7  =  $1,925.     Ans. 

To  find  the  cost  price  when  the  selling  price  and  profit  per  cent,  are 

given. 
s  =  c-\-cr  =  c  (1+r)  .  •  .  c  =zs  ~  (1  -J-  ?•)  =:  100  8  -f-  (100  -f-;>). 

Example.  —  What  cost  that  article  whose  selling  price,  $4,  is 
long  25  per  cent,  of  the  cost  ?  What  price,  more  25  per  cent,  of 
it,  is  equal  to  S4  ?  $4  is  the  sum  of  what  price  and  25  per  cent, 
of  it?     400 -j- 125  =  $3.20.     Ans. 

To  find  the  cost  price  when  the  selling  price  and  loss  per  cent,  are 

given. 

sr=c  —  cr  =  c(l — r)  .-.  c  =  5~(l— 7-)  =  100s-f-(100 — p) 

Example.  —  What  cost  that  article  whose  selling  price,  $375, 
is  short  7  per  cent,  of  the  cost  ?     What  price  less  7  per  cent,  of  it 
is  equal  to  $375  ?       $375  is  the  difference  of  what  price  and  7  per 
cent,  of  it  ? 
375  -J-  (1  —  .07)  =  375  -^-  .93  =  375  X  100  -^-  (100  —  7)  = 

$403,226.     Ans. 


EQUATION  OF  PAYMENTS,  OR  AVERAGE. 

Average  consists  in  finding  the  time  at  which  several  sums, 
foiling  due  at  different  dates,  become  due  if  taken  collectively. 

Rule.  —  Multiply  each  sum  respectively  by  the  number  of 
days  it  falls  due  later  than  that  failing  due  at  the  earliest  date,  and 
divide  the  sum  of  the  products  by  the  sum  of  the  several  sums. 
The  quotient  will  be  the  number  of  days  subsequent  to  the  earliest 
date  at  which  the  whole  will  mature,  or  averages  due. 

.  :  .—  .w  i  i:  \<.i.  gtn  - 1  no  '•  interest  on  i"t<  >■<.<("  to  the  creditor.    l\ 
not  give  1 1  i  in  his  just  due.    It  estimates  by  waj  of  the  fnft  /•«  ri  on  both  ride*,  on 
the  Minis  falling  due  prior  to  the  average  date,  and  on  those  falling  due  - 
quently,  and  not  by  the  Interest  <>n  those  (ailing  due  prior,  and  by  the  dUotmnt 
on  those  (ailing due  subsequent,  as  would  be  strict iv  correct.    The  praetl 
against  the  creditor  or  holder  of  the  demands,  in  like  manner  and  relati 
tent,  as  shown  in  note  under  DlW  oi  N  >. 


EQUATION   OF    PAYMENTS.  I:;;; 

The  following  exhibits  the  face  of  an  account  in  the  ledger,  and 
the  time  (date)  at  which  it  averages  due  is  required. 


36C 

),  April  10 $250.26  —  6  mo. 

Due  Oct. 

10. 

u 

June   25 320.56  —  6    " 

"    Dec. 

25. 

it 

July    10 50.02  —  3    " 

"   Oct. 

10. 

u 

Aug.     1 210.84  —  4    " 

u   Dec. 

1. 

(( 

"      18 73.40  —  5    " 

"   Jan. 

18. 

(( 

Oct.     15 100.      —  cash* 

"   Oct. 

15. 

Example.  —  Practical  method  of  stating  and  working. 
1860.  Due  Oct.  10,  $301 

"        "    Dec.25,     321  X    76  =■  24396. 

"    ■    "       «       1,    211  X    52=  10972. 

"        "    Jan.  18,      73  X  100  =    7300. 

"        "    Oct.   15,    100  X      5  a      500. 

1006  )  43168  (  43  days,  =  Nov.  22,  1860. 

Ans. 
COMPOUND    AVERAGE. 

Compound  Average  consists  in  finding  the  time  at  which  the  bal- 
ance of  an  account  or  demand  averages  due,  whose  sides  —  the  debit 
and  the  credit  —  average  due  at  different  dates. 

Rule.  —  Multiply  the  less  sum  or  side  by  the  difference  in  days 
between  the  two  dates  —  that  at  which  the  debit  side  averages  due 
and  that  at  which  the  credit  side  averages  due  —  and  divide  the  prod- 
uct by  the  difference  of  the  sums  or  sides ;  the  quotient  will  be  the 
number  of  days  that  one  of  the  dates  must  be  set  back,  or  the  other 
forward,  to  mark  the  time  sought ;  for  which  last, 

SPECIAL    RULE. 

Earlier  date  with  larger  sum,  set  back  from  earlier. 
Later,  date  with  larger  sum,  set  forward  from  later. 

Example. — The  debit  side  of  an  account  in  the  ledger  foots  up 
$400,  and  averages  due  Oct.  12,  1860 ;  the  credit  side  of  the  same 
account  foots  $300,  and  averages  due  Nov.  16,  1860.     At  what  date 
does  the  balance  or  difference  between  the  two  sides  average  due  1 
400  300 

300  35 

"TOO    )  10500  (  105  days  earlier  than  Oct.  12,  =  June  29, 1860.    Ans. 

Example. — The  debit  side  of  an  obligation  foots  $250,  and  aver- 
ages due  May  17,  1860 ;  the  credit  side  of  the  same  obligation  foots 
$175,  and  averages  due  May  1,  1860.     At  what  date  does  the  differ- 
ence of  the  sides  average  due  1 
250  175 

175  _J6 

75      )  2800  (  37J  days  later  than  May  17,  =  June  23,  1860.  Ans. 
12 


134  GENERAL   AVERAGE. 


GENERAL  AVERAGE. 

It  is  the  established  usage  that  whatever  of  either  of  the  three 
commercial  interests  —  the  ship,  the  cargo,  or  the  freight  —  is 
voluntarily  sacrificed  or  destroyed  for  the  general  good,  or  with 
the  view  of  saving  the  most  that  may  be  saved  when  all  is  in  immi- 
nent danger  of  being  lost,  is  matter  of  general  loss  to  the  respec- 
tive interests,  and  not  more  especially  to  the  interest  voluntarily 
abandoned  than  to  the  others.  So,  too,  the  losses  and  damages  inci- 
dent to  the  voluntary  sacrifice,  and  collateral  therewith,  together 
with  the  expenditures  which  the  master  has  been  compelled  to 
make  for  the  general  good,  in  consequence  of  disaster,  are  matters 
of  general  average,  or  are  to  be  contributed  for,  pro  rata,  by  the 
several  interests. 

The  contributory  interests  are  the  ship,  the  cargo,  and  the 
freight,  at  their  net  values,  independent  of  charges,  premiums 
paid  for  insurance,  &c. 

The  contributory  value  of  the  ship,  generally,  is  her  value  at  the 
port  of  departure  at  the  time  of  leaving,  less  the  premium  paid  for 
her  insurance. 

The  contributory  value  of  the  cargo  is  its  net  value,  in  a  sound 
state,  at  the  port  of  destination,  if  the  voyage  be  completed ;  or  its 
invoice  value  if  the  voyage  be  broken  up  and  the  cargo  returned 
to  the  port  whence  it  was  shipped  ;  or  its  market-value  at  any  in- 
termediate port,  where  of  necessity  it  is  discharged  and  disposed  of. 
The  value  of  the  goods  jettisoned,  and  to  be  contributed  for,  is 
their  value  after  the  same  manner ;  and  that  value  is  a  part  of  the 
contributory  value  of  the  cargo,  as  well  as  a  matter  of  general 
average. 

The  contributory  value  of  the  freight,  generally,  is  the  gross 
amount  or  amount  per  freight-list,  less  one-third  part  thereof,  in 
most  of  the  States ;  but,  in  the  State  of  New  York,  one-half  thereof, 
for  seamen's  wages  and  other  expenses.  The  loss  of  freight  by 
"jettison,  when  any  freight  is  earned,  is  matter  of  general  average. 
If  the  cargo  is  transshipped  on  board  another  vessel,  and  in  that 
way  sent  to  the  port  of  destination,  the  contributory  value  of  the 
freight  is  the  gross  amount,  less  the  sum  paid  the  other  vessel. 

The  voluntary  damage  to  the  ship,  with  a  view  to  the  general 
good,  —  such  as  throwing  over  her  furniture,  destroying  bet  equip- 
ments, cutting  away  her  masts,  breaking  up  her  decks  to  Lr«t  .it  toe 
cargo  for  the  purpose  of  throwing  it  over,  &c,  —  is  contributed  for 
at  two-thirds  the  cost  of  repairing  and  restoring  ;  the  new  articles 
being  supposed  one-half  better,  or  worth  one-half  more,  than  the 
old. 


GENERAL   AVERAGE.  135 

If  we  let  V  =  contributory  value  of  the  vessel, 
C  =  contributory  value  of  the  cargo, 
F  =  contributory  value  of  the  freight, 
d  z=.  aggregate  amount  of  losses  to  be  averaged,  then 
d  -J-  ( V  -f-  C  -f-  F)  =  r,  the  per  cent,  of  each  interest  that  each 
must  contribute,  and 

VX  r  =  Vessel's  share  of  the  loss, 
C  X  t  ■=.  Cargo's  share  of  the  loss, 
F  X  t  =  Freight's  share  of  the  loss. 
When  a  contributory  interest's  share  of  the  loss  is  to  be  distrib- 
uted among  the  several  owners  of  that  interest,  the  same  pro  rata 
method  is  to  be  observed  :  thus 

A  X  r  =  sum  A-  must  contribute, 
B  X  r  =  sum  B  must  contribute, 
D  X  t  =  sum  D  must  contribute  ; 
A,  B,  and  D  being  A's,  B's,  and  D's  respective  shares  in  that 
interest. 


136  ASSESSMENT   OF   TAXES. — INSURANCE. 


ASSESSMENT  OF  TAXES. 

G  =  amount  of  taxable   property,  real   and  personal,  as  per 
grand  list. 

A  =  amount  of  money  to  be  raised,  including  the  whole  poll-tax. 

T  =  amount  of  money  to  be  raised  on  property  alone. 

n  =  number  of  ratable  polls. 

h  =  poll-tax  per  head. 

r  =  rate  per  cent,  to  be  raised  on  taxable  property. 

P  =  an  individual's  taxable  property,  as  per  grand  list. 

b  =  P's  poll-tax. 

Tz=A  —  hn.     r  =  T  -f-  G.     P  r  -f  b  =  Fs  tax,  including  poll. 


INSURANCE. 

Insurance  is  a  written  contract  of  indemnity,  called  the  policy, 
by  which  one  party  (the  insurer  or  underwriter)  engages,  for  a 
stipulated  sum,  called  the  premium  (usually  a  per  cent,  on  the 
value  of  the  property  insured),  to  insure  another  against  a  risk  or 
loss  to  which  he  is  exposed. 

Let  P=  Principal,  or  amount  insured  on, 
r  =  rate  per  cent,  of  insurance, 
a  =  premium  for  insurance. 

a  =  Pr.    r  =  a  -f-  P.     P  =  a  -f-  r. 

Example. — What  is  the  premium  for  insuring  on  &4500  at  lj 
per  cent.  ? 

4500  X  -015  =  $67.50.     Ans. 


LIFE-INSURANCE. 

Life-insurance  is  predicated  upon  the  even  chance  in  years, 
called  the  expectation  of  life,  that  an  individual  in  general  health 
at  any  given  age  appears  by  the  rates  of  mortality  to  have  of  living 
beyond  that  age. 

The  Carlisle  Tables  of  Expectation,  column  C  in  the  following 
tables,  are  used  almost  or  quite  exclusively  in  England,  and  by 
some  insurance-companies  in  the  United  States;  while  tli<»e  by 
Dr.  Wiggleiwtiirth,  column  \V,  computed  with  special  reference  to 
the  rates  of  mortality  in  this  country,  arc  used  by  others. 

The  Supreme  Court  of  Massachusetts  has  adopted  the  Wiggles- 


worth  rates  of  expectation  in  estimating  the  value  of  life-annuities 
and  life-estates. 

TABLE 

Of  Ages  and  Expectations  jrom  Birth  to  103  Years. 


Age. 

C. 

W. 

Age. 

0. 

w. 

Age. 
52 

c. 

w. 

0. 

w. 
6.59 

0 

38.72 

28.15 

26 

87.14 

31.93 

19.68 

20.05 

78 

6.12 

1 

44.68 

27 

36.41 

31.50 

53 

18.97 

L9.46 

79 

6.21 

2 

47.55 

88.74 

28 

35.69 

31.08 

54 

18.28 

L8.92 

80 

5.51 

3 

40.01 

29 

35.0030.66 

55 

l  7.58 

18.35 

81 

5.21 

5.50 

4 

50.76 

10.79 

30 

30.25 

56 

16.89 

17.78 

82 

4.93 

5.16 

5 

51.25 

40.88; 

31 

33.68 

29.88 

57 

L6.2J 

17.20 

83 

4.65 

4.S7 

6 

51.17 

40.69 

32 

83.03 

29.48, 

58 

15.55 

16.68 

84 

4.39 

1.66 

7 

40.47 

33 

82.86 

29.02 

59 

14.92 

L6.04 

85 

4.12 

4.57 

8 

50.84 

40.14 

34 

31.68 

28.62 

60 

14.34 

15.45 

'86 

3.90 

4.21 

9 

19.57 

39.72 

35 

31.00 

28.22 

61 

13.82 

14.86 

87 

8.71 

8.90 

10 

48.82 

39.23 

36 

30.32 

27.78 

62 

13.81 

14.26 

88 

3.59 

3.67 

11 

48.04 

38.64 

37 

29.64 

27.34 

63 

12.81 

13.66 

89 

3.47 

8.66 

12 

47.27 

38.02 

38 

28.96 

26.91 

64 

12.30 

13.05 

90 

8.28 

3.43 

13 

46.51 

37.41 

39 

28.28 

26.4  7 

65 

11.79 

12.43 

91 

3.26 

3.32 

14 

i. ->.:;. 

36.79 

40 

27.61 

26.04 

66 

11.27 

11.96 

92 

3.37 

3.12 

15 

45.00 

36.17 

41 

26.97 

25.61 

67 

ID.  7.") 

11.48 

93 

3.48 

2.40 

16 

44.27 

35.76 

42 

26.34 

25.19 

68 

10.23 

11.01 

94 

3.53 

1.98 

17 

43.57 

35.37 

43 

25.71 

21.77 

69 

9.70 

10.50 

95 

3.53 

1.62 

18 

42.87 

34.98 

44 

25.09 

24.35 

70 

9.18 

10.06 

96 

3.46 

19 

42.17 

34.59 

45 

24.46 

23.92 

71 

8.65 

9.60 

97 

3.28 

20 

41.46 

34.22 

46 

23.82 

23.37 

72 

8.16 

9.14 

98 

3.07 

21 

40.75 

33.84 

47 

23.17 

22.83 

73 

7.72 

8.69 

99 

2.77 

22 

40.04 

33.46 

48 

22.50 

22.27 

74 

7.33 

8.25 

100 

2.28 

23 

39.31 

33.08 

49 

21.81 

21.72 

75 

7.01 

7.83 

101 

1.79 

24 

38.59 

32.70 

50 

21.11 

21.17 

76 

6.69 

7.40 

102 

1.30 

25 

37.86 

32.33 

51 

20.39 

20.61 

77 

6.40 

6.99 

103 

0.83 

Thus,  by  the  tables,  a  man  in  general  good  health  at  21  years  of 
age  has  an  even  chance,  by  the  Carlisle  rate  of  mortality,  of  living 
40!  years  longer;  by  the  Wigglesworth  rate,  of  living  33^- 
years  longer.  So  a  man  in  general  good  health,  at  60  years  of 
age,  has,  by  the  Carlisle  rate,  an  even  chance  of  living  14.34  years 
longer;  by  the  Wigglesworth  rate,  an  even  chance  of  living  15.45 
years  longer,  etc. 
12* 


138 


FELLOWSHIP. 


FELLOWSHIP. 


Fellowship  calls  for  the  distribution  of  a  given  effect  to  each 
of  the  several  causes  associated  in  its  production,  proportional  to  their 
respective  magnitudes  one  with  another. 

It  is  a  rule,  therefore,  adapted  to  the  use  of  partners  associated  in 
business,  in  achieving  a.  pro  rata  distribution  among  themselves  as  indi- 
viduals, of  the  profits  or  losses  pertaining  to  the  company. 

Rule.  —  Multiply  each  partner's  investment  or  share  of  the  capital 
stock,  by  the  whole  gain  or  loss,  and  divide  the  product  by  the  sum 
of  all  the  shares,  or  gross  capital. 

Example.  —  Three  men,  A,  B,  and  C,  enter  into  partnership.  A 
invests  $500,  B  $700,  and  C  $300.  They  trade  and  gain  $400. 
What  is  each  partner's  share  of  the  profits  * 


A,  $500 

B,  700 

C,  300 


500  X  400  +  1500  =  $133. 33£  =  A's  share. 
700  X  400  -J-  1500  =     I86.66jf  =  B's     " 
300  X  400  -f-  1500  =      80.00    =  C's     " 


$1500  =  gross  capital.  $400.00     Proof. 

Example. — D's  investment  of  $600  has  been  employed  eight 
months ;  E's,  of  $500,  five  months  ;  and  F's,  of  $300,  five  months  ; 
the  profits  of  the  company  are  $500,  and  are  to  be  divided  pro  rata 
among  the  partners.     What  is  each  partner's  share  ? 

D,  $600  X  8  =  4800  X  500  -r-  8800  =  $272.73,  D's  share. 

E,  500  X  5  =  2500  X  500  -f-  8800  =     142.05,  E's      " 

F,  300  X  5  =  1500  X  500  -r-  8800  =       85.22,  F's      " 

8800  $500.    Proof. 

Example.  —  Of  $120  distributed,  there  were  given  to  A,  J  ;  to  B, 
£  ;  to  C,  £ ;  and  to  D,  £,  and  there  was  nothing  remaining. 
What  sum  did  each  receive  1 

J  of  120  =  40  X  120  -T-  114  =  $42T\  =  A's  share, 
j  of  120  =  30  X  120 -r-  114=    31-f^  =  B's       " 
£  of  120  =  21  X  120  -h  114  =    25^9  =  C's       " 
I  of  120  =  20  X  120  -J-  114  =    21^  =  D'a      " 
Til  $120.     Proof. 

Example.  —  Divide  the  number  180  into  3  parts,  which  shall 
be  to  each  other  as  2,  3,  4. 

J  of  180  =  90  X  180  4-195  =  83.08 
of  180  =  GO  X  180  -f-  195  =  55.88 
of  180  =  45  X  180  4-195  =  41.54 

195  180.00     Proof. 


ALLIGATION.  189 

Example. — $400  are  to  be  divided  between  A,  B,  and  C,  in 
the  ratio  of  £  to  A,  £  to  B,  and  |  to  C;  how  much  will  each 
receivg  ? 

4  of  400  =  200,  and  200  X  400-^-500  =  $160  =  A's  share. 

X  of  400  =  200,  and  200  X  400  -^-  500  =    160  =  B's  share. 

J  of  400  =  100,  and  100  X  400  -f-  500  =      80  =  C's  share. 

500  $400.     Proof. 


ALLIGATION. 

Alligation  Medial  is  a  method  by  which  to  find  the  mean  price  of 
a  mixture  or  compound,  consisting  of  two  or  more  articles  or  ingre- 
dients, the  quantity  and  price  of  each  being  given. 

Rule.  —  Multiply  each  quantity  by  its  price,  and  divide  the  sum  of 
the  products  by  the  sum  of  the  quantities ;  the  quotient  will  be  the 
price  per  unity  of  measure  of  the  mixture ;  and,  having  found  tha 
price  of  the  given  quantities  as  mixed,  any  quantities  of  the  same 
materials,  taken  in  like  proportions,  will  be  at  the  same  price. 

Example.  — If  20  lbs.  of  sugar  at  8  cents,  40  lbs.  at  7  cents,  and 
80  lbs.  at  5  cents  per  pound,  be  mixed  together,  what  will  be  the  mean 
price,  or  price  per  pound,  of  the  mixture? 
20  X  8  =  160 

40X7  =  280 
80X5  =  400 

140  )  840  (  6  cents.     Ans. 

The  several  kinds,  then,  at  their  respective  prices,  taken  in  the 
proportion  of  1  at  8,  2  at  7,  and  4  at  5  cts.,  will  form  a  mixture  worth 
6  cts.  a  pound. 

Example.  —  If  10  lbs.  of  nickel  are  worth  $2,  and  24  lbs.  of  copper 
are  worth  $4£,,and  8  lbs.  of  zinc  are  worth  40  cts.,  and  1  lb.  of  lead  is 
worth  5  cts.,  what  are  5  lbs.  of  pretty  good  German  silver  worth? 
(iiUL±JLiJ^+4JL±_a)Ki  =  81  cents.     Ans. 

Alligation  Alternate  is  a  method  by  which  to  find  what  quantity 
of  each  of  two  or  more  articles  or  ingredients,  whose  prices  or  quali- 
ties are  given,  must  be  taken  to  form  a  mixture  or  compound  that 
shall  be  at  a  given  price  or  of  a  given  quality  between  the  two 
extremes.  It  also'applies  to  the  finding  of  relative  quantities  when 
the  quantity  of  one  or  more  of  the  articles  is  limited. 

Rule.  —  Connect  the  given  prices  or  qualities  —  a  less  than  the 
given  mean  with  that  one  or  either  one  that  is  greater  —  and  to  the 
extent  that  all  be  thus  connected ;  then  plaee  the  difference  between 


140 


ALLIGATION. 


each  given  and  the  given  mean  opposite,  not  the  given,  or  the 
given  mean,  but  the  given  with  which  it  is  alligated ;  the  num- 
ber standing  opposite  each  price  or  quality  will  be  the  quantity  that 
must  be  taken  at  that  price,  or  of  that  quality,  to  form  a  mixture  or 
compound  at  the  price  or  of  the  quality  desired.  And,  being  propor- 
tions respectively  to  each  other,  they  may  be  taken  in  ratio  greater 
or  less,  as  desired. 

Example.  —  In  what  proportions  shall  I  mix  teas  at  48  cents  a 
pound  and  54  cents  a  pound,  that  the  mean  price  may  be  50  cents  a 
pound? 

In  the  proportions 

,A  5  48i  (  4  lbs  at  48  cts.    )    A 
50  >  54J  )  2  lbs.  at  54  cts.  J^15' 
Or,  as  2  at  48  to  1  at  54. 


Proof.  5  2  X  48  -f  1  X  54  =  150. 


3  X  50 


=  150. 


Example.  —  In  what  proportions  shall  I  mix  teas  at  48,  54,  and  72 
cents  a  pound,  that  the  mixture  may  average  60  cents  a  pound  ? 
(481       12,  12  at  48)        ( 2  at  48 

60  {  54-||       12,  12  at  54  >  =  <  2  at  54  V 


72J 


18 


y  avcidgc  uu  i.cuis  <i  jr 

at  48  )  (  2  at  48  ) 
at  54  }  =  <  2  at  54  V 
at  72  )        (  3  at  72  J 


12  +  6, 

Example.  —  A  wine  dealer  has  received  an  order  for  a  quantity  of 
wine  at  50  cts.  a  gallon.  He  has  none  ready  manufactured  at  that 
price.  He  has  it  at  40  cts.,  at  56  cts.,  and  at  80  cents  a  gallon,  and 
he  has  water  that  cost  him  nothing.  He  wishes  to  fill  the  order 
with  a  mixture  composed  of  the  four  materials  —  the  water  and  the 
three  different  priced  wines.  In  what  proportions  must  he  mix  them, 
that  the  mean  or  average  price  may  be  50  cents? 

Ans. 

30  =  301 

6  +  30  =  36  I 

10  =10  f 

80=^50 -{-10  =  60  J 


Or,  50 


176  gals. 


66J 
80- 

=  136  gals. 
Ans. 

6            1 

|    6  4-  30  1 

50  -f  10  f 
J 10           J 

=  1 12  gals. 

If,  now,  having  found  the  proportions  desired,  it  is  wished  to  limit 
\e  of  the  articles  in  quantity  —  say  the  best  wine  to  8  gallons  in  the 


INVOLUTION  —  EVOLUTION.  141 

mixture  —  the  pioportions  of  the  remaining  articles  thereto  are  found 
thus  :  — 

Instance,  1st  example, — 

10 

10 

10 


;,  1st  example,  — 

8  ::  50  =10    )   .     .    .  .„  .       ~ 

o  ..  on o  i    (And  the  mixture  will  consist  of 

8  ::    6=  44  (  8  +  40  +  24  +  4t  =  76t  Sallons- 


If,  instead,  it  is  desired  to  mix  a  given  quantity,  say  100  gallons, 
and  proportioned,  say  as  in  first  example,  the  quantity  to  be  taken  of 

each  is  ascertained  by  the  following 

Rule.  —  As  the  sum  of  the  relative  quantities  is  to  the  quantity 
required,  so  is  each  relative  quantity  to  the  quantity  required  of  it 
respectively. 

The  sum  of  the  relative  quantities  alluded  to  is  6  +  30  -J-  50  -\- 10 
—  96;  then, 

96  :  100  ::  6  =  6| 
y«  :  100  ::  30=314 
96  :  100  ::  50  =  52^ 
96  :  ioo  ::  io=ioT*2 


INVOLUTION. 

Involution  consists  in  involving,  that  is,  in  multiplying  a  number 
one  or  more  tines  into  itself.  The  number  so  involved  is  called  the 
root,  and  the  produet  arising  from  such  involution,  its  power. 

The  second  power,  or  square,  of  the  root,  is  obtained  by  multiplying 
the  root  once  into  itself,  as4  X  4=»16;  4  being  the  root  and  16  its 
square. 

The  third  poiver,  or  cube,  of  a  number,  is  obtained  by  multiplying 
the  number  twice  into  itself,  as4  X  4X  4  =  64;  and  so  on  for  any 
power  whatever. 

When  a  number  is  to  be  involved  into  itself,  a  small  figure  called 
the  index  or  exponent  is  placed  at  its  right,  indicating  the  number  of 
times  it  is  to  be  so  involved,  or  the  power  to  which  it  is  to  be  raised. 
Thus,  31  =  3  X  3  X  3  X  3  =  81;  and43  =  4  X  4  X  4  =  64. 


EVOLUTION. 

Evolution  is  the  opposite  of  Involution.  It  consists  in  finding  a 
toot  of  a  given  number,  instead  of  a  power  of  a  given  root. 

When  the  root  of  a  number  is  required  or  indicated,  the  number  is 
A'ritten  witb  the  V  before  it  :  and  the  character  or  denomination  of 
the  root,  if  it  be  other  than  the  square  root,  is  defined  by  an  index 


142  EVOLUTION. 

figure  placed  over  the  sign.  When  the  square  root  of  a  number  is 
required,  the  sign  (V)  is  placed  before  the  number,  but  the  index 
(2)  is  usually  omitted.  Thus,  V25,  shows  that  the  square  root  of 
25  is  required,  or  to  be  taken  ;  and  /^25  shows  that  the  cube  root 
is  required.     The  operation  is  usually  called  extracting  the  root. 

TO  EXTRACT  THE  SQUARE  ROOT. 

Rule  —  1.  Separate  the  given  number  into  periods  of  two  figures 
each,  by  placing  a  point  over  the  first  figure,  third,  fifth,  &c,  counting 
from  right  to  left — the  root  will  consist  of  as  many  figures  as  there 
are  periods. 

2.  Find  the  greatest  square  in  the  left  hand  period,  and  place  its 
root  in  the  quotient ;  subtract  the  square  of  the  root  from  the  left 
hand  period,  and  to  the  remainder  bring  down  the  next  period  for  a 
dividend. 

3.  Multiply  the  root  so  far  found  —  the  figure  in  the  quotient  —  by 
2,  for  a  divisor ;  see  how  many  times  the  divisor  is  contained  in  the 
divided,  except  the  right  hand  figure,  and  place  the  result  (the  num- 
ber of  times  it  is  contained)  in  the  quotient,  to  the  right  of  the  figure 
already  there,  and  also  to  the  right  of  the  divisor  ;  multiply  the  divi- 
sor, thus  increased,  by  the  last  figure  in  the  quotient,  and  subtract  the 
product  from  the  dividend,  and  to  the  remainder  bring  down  the  next 
period  for  a  dividend. 

4.  Multiply  the  quotient  —  the  root  so  far  found  (now  consisting  of 
two  figures)  —  by  2,  as  before,  and  take  the  product  for  a  divisor ; 
see  how  many  times  the  divisor  is  contained  in  the  dividend,  except 
the  right  hand  figure,  and  place  the  result  in  the  quotient,  and  to  the 
right  of  the  divisor,  as  before  ;  multiply  the  divisor,  as  it  now  stands, 
by  the  figure  last  placed  in  the  quotient,  and  subtract  the  product  from 
the  dividend,  and  to  the  remainder  bring  down  the  next  period  for  a 
dividend,  as  before. 

5.  Multiply  the  quotient  (now  consisting  of  3  figures)  by  2,  as 
before,  and  take  the  product  for  a  divisor,  and  in  all  respects  proceed 
as  when  seeking  for  the  last  two  figures  in  the  quotient.  The  quo- 
tient, when  all  the  periods  have  been  brought  down  and  divided,  will 
be  the  root  sought. 

Notb.  —  1.  If  there  is  a  remainder  after  finding  the  integer  of  a  root,  annex  periods  of 
ciphers  thereto,  and  proceed  as  when  seeking  for  the  integer.  The  quotient  figures  will 
be  the  decimal  portion  of  the  root. 

2.  If  the  given  number  is  a  decimal,  or  consists  of  a  whole  number  and  decimal,  point 
off  the  decimal  from  left  to  right,  by  placing  the  point  over  the  secomi,  fourth,  sixth,  &.c, 
figures  therein,  aiwl  fill  the  last  period,  if  incomplete,  bjf  ipher. 

3.  If  the  dividend  does  not  contain  the  divisor,  a  Cipher  must  he  placed  in  the  quotient, 
and  also  at  the  right  of  the  divisor,  and  the  next  period  brought  down ;  then  the  dividend 
must  be  divided  by  the  divisor  as  increased. 

4.  If  the  quotient  figure,  obtained  by  dividing  by  the  double  of  the  root,  is  too  large,  as 
will  sometimes  be  the  case,  (see  3d  Example)  it  must  be  dropped,  and  a  lees  — one  which 
m  the  truo  measure  —  taken  in  its  stead. 


EVOLUTION.  143 


Example.  —  Required  the  square  root  of  123456.432. 

123456.4320  (  351.3636-f-.  Ans. 
9 

65  )  334 
325 


70  .  )  956 
701 


7023  )  25543 
21069 


70266  )  447420 
421596 

702723  )  2582400 
2108169 


7027266  )  47423100 
42163596 

5259504 

Example.  —  Required  the  square  root  of  10621.     Also,  of  28561. 


10621  (  103.05+.     Ans. 

1 


203  )  00621 
609 
20605  )  120000 
103025 
16975 


28561  (  169.  Ans. 

1 


26  )  185 
156 


329  )  2961 
2961 


TO    EXTRACT    THE    CUBE    ROOT. 

Rule —  1.  Separate  the  given  number  into  periods  of  three  figures 
each,  by  placing  a  point  over  the  first,  fourth, seventh, &c,  counting 
from  right  to  left — the  root  will  consist  of  as  many  figures  as  there 
are  periods. 

2.  Find  the  greatest  cube  in  the  left  hand  period,  and  place  its  root 
in  the  quotient ;  subtract  the  cube  of  the  root  from  the  left  hand  pe- 
riod, and  to  the  remainder  bring  down  the  next  period  for  a  dividend. 

3.  Multiply  the  square  of  the  quotient  by  300,  for  a  divisor ;  see 
how  many  times  the  divisor  is  contained  in  the  dividend,  and  place 
the  result  (except  that  the  remainder  is  large,  diminished  by  one  or 
two  units)  in  the  quotient. 

4.  Multiply  the  divisor  by  the  figure  last  placed  in  the  quotient, 
and  to  the  product  add  the  square  of  the  same  figure,  multiplied  by  the 
other  figure,  or  figures,  in  the  quotient,  and  by  30  ;  and  add  also  thereto 


144  EVOLUTION. 

the  cube  of  the  same  figure,  and  take  the  sum  for  the  subtrahend ;  sufr. 
tract  the  subtrahend  from  the  dividend,  and  to  the  remainder  bring 
down  the  next  period  for  a  dividend,  with  which  proceed  as  with  the 
preceding,  so  continuing  until  the  whole  is  completed. 

Note  — 1.  Decimals  must  be  pointed  from  left  to  right,  by  placing  a  point  over  the 
third,  sixth,  &c,  figures  in  that  direction. 

2.  If  the  divisor  is  not  contained  by  the  dividend,  place  a  cipher  in  the  quotient,  and 
annex  two  ciphers  to  the  divisor,  and  bring  down  the  next  period  for  a  dividend,  and  use 
the  divisor,  as  thus  increased,  for  finding  the  next  quotient  figure. 

3.  If  there  is  a  remainder  after  finding  the  integer  of  the  root,  annex  a  period  of  three 
ciphers  thereto,  and  proceed  for  the  decimal  of  the  root  as  if  seeking  for  the  integer,  an- 
nexing a  period  of  three  ciphers  to  each  remainder  until  the  decimal  is  carried  to  as  many 
places  of  figures  as  desired. 

Example.  —Required  the  cube  root  of  47421875.6324. 

4742i875.632400  (  361.959-J- . 
27  Am. 

32X  300  =  2700)  20421 
6 

16200 
6s   X  3  X  30      =    3240 

63       =     216=19656 


362  X  300  =  388800  )  765875 
1 


388800 
l2  X  36  X  30   =  1080 

I3  =    1  =  389881 

36 12  X  300  =  390963Q0  )  375994632 
9 


351866700 
B2  X  361  X  30  =   877230 

93=     729  =  352744659 

36192  X  300  =  3929148300  )  23249973400 
5 


19615711500 
5s  X  3619  X  30  =     2714250 

5^= 125  =  19648455875 

361952  X  300  =  393023107500  )  3601517525000 
9 


3537210667500 
S*  X  36195  X  30  a  87953850 
03  =- 729  »  3537298622079 

64218902921 


EVOLUTION.  145 

Example.  -  -  Required  the  cube  root  of  32768.     Also,  of  8489664. 


32768(32. 
27 Ans. 

32X300  =  2700  )  5768 
2 

5400 
22X  3X30=360 

23=    8  =  5768 


8489664(204. 
8  Ans. 


2s  X  300  =  120000  )  489664 
4 

480000 
42X  20X30    =9600 

43  =      64=480664 


General  Rule  for  extracting  the  roots  of  all  powers,  or  for  finding 
any  proposed  root  of  a  given  number. 

1.  Point  off  the  given  number  into  periods  of  as  many  figures 
each,  counting  from  right  to  left,  as  correspond  with  the  denomina- 
tion of  the  root  required ;  that  is,  if  the  cube  root  be  required,  into 
periods  of  three  figures,  if  the  fourth  root,  into  periods  of  four  fig- 
ures, &c. 

2.  find  the  first  figure  of  the  root  by  inspection  or  trial,  and  place 
it  at  the  right  of  the  number,  in  the  form  of  a  quotient ;  raise  this 
quotient  figure  to  a  power  corresponding  with  the  denomination  of 
the  root  sought,  and  subtract  that  power  from  the  left  hand  period, 
and  to  the  remainder  bring  down  the  first  figure  of  the  next  period, 
for  a  dividend. 

3.  Raise  the  root  thus  far  found  (the  quotient  figure)  to  a  power 
next  inferior  in  denomination  to  that  of  the  root  required,  multiply 
this  power  by  the  number  or  index  figure  of  the  root  required,  and 
take  the  product  for  a  divisor ;  find  the  number  of  times  the  divisor 
is  contained  in  the  dividend,  and  place  the  result  (except  that  the 
remainder  is  large,  diminished  by  one  or  two  units)  in  the  quotient, 
for  the  second  figure  of  the  root. 

4.  Raise  the  root  thus  far  found  (now  consisting  of  two  figures)  to 
a  power  corresponding  in  denomination  with  the  root  required,  and 
subtract  that  power  from  the  two  left  hand  periods,  and  to  the  re- 
mainder bring  down  the  first  figure  of  the  third  period,  for  a  divi- 
dend ;  find  a  new  divisor,  as  before,  and  so  proceed  until  the  whole 
root  is  extracted. 

Example.  —  Required  the  fifth  root  of  45435424. 

45435424(34.    Ans. 
35  =  243. 

3^X5)2113 
345=45435424 

13 


146  AKITHMETICAL   PROGRESSION. 

Example.  — Required  the  fifth  root  of  432040.0o54. 

432040.03540  (  13.4  -f.     Arts. 

14X5)33 
135  =  371293 


134X5)  607470 
13.45  =  43204003424 

116 

For  instructions  touching  special  cases,  see  Notes  relative  to  the 
extraction  of  the  square  root,  and  to  the  extraction  of  the  cube  root. 

The  a/  of  the  a/  of  any  number  =  /s/  of  that  number 
"     Vofthe^  =  /y. 
"     V  of  the  V  of  the  s/  =  £/ . 
"     #  of  the  a/ =  a/. 
"    V  of  the  &  =&,  &c. 


ARITHMETICAL  PROGRESSION. 

A  series  of  three  or  more  numbers,  increasing  or  decreasing  by 
equal  differences,  is  called  an  arithmetical  progression.  If  the  num- 
bers progressively  increase,  the  series  is  called  an  ascending  arith- 
metical progression ;  and  if  they  progressively  decrease,  the  series  is 
called  a  descending  arithmetical  progression. 

The  numbers  forming  the  series  are  called  the  terms  of  the 
progression,  of  which  the  first  and  the  last  are  called  the  extremes, 
and  the  others  the  means. 

The  difference  between  the  consecutive  terms,  or  that  quantity  by 
which  the  numbers  respectively  increase  upon  each  other,  or  decrcaso 
from  each  other,  is  called  the  common  difference. 

Thus,  3,  5,  7,  9,  11,  *C.,  is  an  ascending  arithmetical  progression, 
and  11,  9,  7,  5, 3,  is  a  descending  arithmetical  progression.  In  these 
professions,  in  both  instances,  11  and  3  are  the  extremes,  of  whick 
11  is  the  greater  extreme,  and  3  is  the  less  extreme.  The  Humbert 
between  these,  (9,  7,  5,)  arc  the  means. 

In  every  arithmetical  progression,  the  sum  of  the  extremes  is 
equal  to  the  sum  of  any  two  moans  that  arc  equally  distant  from  the 
extremes;  and  is,  therefore,  equal  to  twice  the  middle  term,  \v lion 
the  series  consists  of  an  odd  Dumber  of  terms.  Thus,  in  the  fore- 
going series,  3  +  11  =  5  -f  9  =  7  X  2. 

The  greater  extreme,  the  less  extreme,  the  nund>er  of  terms,  the 


ARITHMETICAL    PROGRESSION.  147 

common  difference,  and  the  sum  of  the  terms,  are  called  the  Jive  prop- 
erties of  an  arithmetical  progression,  of  which,  any  three  being  given, 
the  other  two  may  be  found. 

Let  s  represent  the  sum  of  the  terms. 
'  '    E       "         the  greater  extreme. 
"    e        "         the  less  extreme. 
lt    d       "        the  common  difference. 
"    n       "        the  number  of*  terms. 

The  extremes  of  an  arithmetical  progression  and  the  number  of  terms 
being  given,  to  find  the  sum  of  the  terms. 

(E  4-  e)  X  n 

2 ■■  8um  °f tne  terms. 

Example.  —  What  is  the  sum  of  all  the  even  numbers  from  2  to 
100,  inclusive  ? 

102  X  50-4-2  =  2550.     Ans. 

Example.  —  How  many  times  does  the  hammer  of  a  common  clock 
strike  in  12  hours  1 

(1  -J-  12)  X  12  -r-  2  =  78  times.    Ans. 
I    ~~e  + 1  j  x     T    =•»  sum  of  tno  terms. 


(E  X  2  —  w—1  X  d)  X  h  n  =  sum  of  the  terms. 


(2e-\~n  —  lX.d)  X  £  w  =  sum  of  the  terms. 

The  greater  extreme,  the  common  difference,  and  the  number  of  terms 
of  an  arithmetical  progression  being  given,  to  find  the  less  extreme. 


E  —  (d  X  n  —  1)  ==  less  extreme. 

Example.  —  A  man  travelled  18  days,  and  every  day  3  miles  far- 
ther than  on  the  preceding ;  on  the  last  day  he  travelled  56  miles ; 
how  many  miles  did  he  travel  the  first  day  ? 


56  —  (18  —  1  X  3)  =  5  miles.     Ans. 

— {  )  =  less  extreme. 

n      V        2        / 

5 

-  X  2  —  E  =  less  extreme. 


148  ARITHMETICAL  PROGRESSION. 

V (EX2  +  d)>  —  s  X  dX$-\-  <*=lesa  extreme,  when 
2 
W  (2  E  -\-.d  )2  —  8  s  d  is  equal  to,  or  greater  than  d. 

a/  {2^t-\-dyi  —  8  sd^d  =  less  extreme,  when 
2 
V (2E-j-rf)2  —  85«? is  less  than  d. 


A/(2e^d)2-±-8sd — <f=»  greater  extreme 
2 


rfX»  —  l-|-e  =  greater  extreme. 


—■  -{- o ■■  greater  extreme. 

25-fn  —  e  =  greater  extreme. 

Tfe  extremes  of  an  arithmetical  progression  and  the  common  difference 
being  given,  to  find  the  member  of  terms. 

E  —  e-r-d-\-l=i  number  of  terms. 

Example.  —  As  a  heavy  body,  falling  freely  through  spaee,  de- 
■cends  16^-  feet  in  the  first  second  of  its  descent,  48^  feet  in  the 
next  second,  80^-  in  the  third  second,  and  so  on ;  how  many  sec- 
onds had  that  body  been  falling,  that  descended  305^.z  fcet  in  the 
last  second  of  its  descent  ? 

305^  —  16^  =  289£  -5-  32£  =  9  +  1  =  10  seconds.     Ans. 

*/(2e^dy  +  8sd  —  d  —  e -e- <* -f  1  =  number  of  terms. 


25-j.E-f  V(2E-f</)t  —  8srf-f  </ 2=  number  of  terms  when 
2 
V  (2  E  -\-  d)2  —  8  5  d  is  equal  to,  or  greater  than  d. 


2  5  -f-  E  -fV(2E-f<f)2  —  Ssd^d  —  number  of  terms  when 
2 

//(2E4-^/)2  —  8*rf  is  less  than  rf. 
*X2 


»+« 


=  number  of  terms. 


ARITHMETICAL   PROGRESSION.  140 

The  extremes  of  an  arithmetical  progression,  and  the  number  of  terms 
being  given,  to  find  the  common  difference, 

E  —  e 

7  =  common  difference. 

n  —  l 

Example.  —  One  of  the  extremes  of  an  arithmetical  progression  is 
28  and  the  other  is  100,  and  there  are  19  terms  in  the  series ;  re- 
quired the  common  difference. 

100^28-j-lool9=a4.     Arts. 


AX  2        \ 

—  e  "5"  \  E4-e  —  *  J  ™  comm011  difference. 


E 

2*-5-n  —  2e 

n  —  1 

2E— (2  5-J-n) 


common  difference. 


=  common  difference, 
n —  I 

Example.  —  The  less  extreme  of  an  arithmetical  progression  is  28, 
the  sum  of  the  terms  1216,  and  the  number  of  terms  19  ;  required 
the  7th  term  in  the  series,  descending.  • 

1216  X  2  -f- 19  =  128  =  sum  of  the  extremes. 
128  —  28  =  100  —  greater  extreme. 
100  —  28  =  72  =  difference  of  extremes. 


72-j-w  —  1  (18)  =  4  =  common  difference. 
100  —  (7  —  1  X  4)  =  76  =  7th  term  descending.   Ans. 
Required  the  5th  term  from  the  less  extreme,  in  an   arithmetical 
progression,  whose  greatest  extreme  is  100,  common  difference  4, 
and  number  of  terms  19. 


100  —  (19  —  5  X  4)  =  44.     Ans. 

To  find  any  Assigned  number  of  arithmetical  means,  between  two  given 
numbers  or  extremes. 

Rule.  —  Subtract  the  less  extreme  from  the  greater,  divide  the 
remainder  by  1  more  than  the  number  of  means  required,  and  the 
quotient  will  be  the  common  difference  between  the  extremes ; 
which,  added  to  the  loss  extreme,  gives  the  least  mean,  and,  added 
to  that,  gives  the  next  greater,  and  so  on. 

Or,  K  —  ~e  -f-  ra  +  1  =  d,  E  being  the  greater  extreme,  e  the  less 
extreme,  m  the  number  of  means  required,  and  d  the  common  differ- 

1  e  -f-  d,  e  -f  2  d,  e+'dd,  &c.  ;  or,  E  —  d,  E  —  2d,  E  —  3  d, 
fcc.,  will  give  the  means  required. 
13* 


150  GJOMETRICAL  PROGRESSION. 

Example. — Required  to  find  5  arithmetical  mean*  between  the 
numbers  18  and  3. 

18  —  3  =  15 H- 6  =  24,  and 
3-^2i=-5i-f.2i=»8-f-24=-104-f2i  =  13-f-24  =  15i. 
5£,  8,  10£,  13, 15£,  therefore,  are  5  arithmetical  means,  between  the 
extremes,  3  and  18. 

Note. — The  arithmetical  mean  between  any  two  numbers  maybe  found  by  dividing 
the  sum  of  those  numbers  by  2  ;  thus,  the  arithmetical  mean  of  0  and  &  is  (&-^-&)-^2=8±. 


GEOMETRICAL  PROGRESSION. 

A  series  of  three  or  more  numbers,  increasing  by  a  common  mul- 
tiplier, or  decreasing  by  a  common  divisor,  is  called  a  geometrical 
progression.  If  the  greater  numbers  of  the  progression  are  to  the 
right,  the  progression  is  called  an  ascending  geometrical  progression, 
but,  on  the  contrary,  if  they  are  to  the  left,  it  is  called  a  descending 
geometrical  progression.  The  number  by  which  the  progression  is 
Formed,  that  is,*the  common  multiplier,  or  divisor,  is  called  the 
ratio. 

The  numbers  forming  the  series  are  called  the  terms  of  the  pro- 
gression, of  which  the  first  and  the  last  are  called  the  extremes,  and 
the  others  the  means.  The  greater  of  the  extremes  is  called  tho 
greater  extreme,  and  the  less  the  less  extreme. 

Thus,  3,  6,  12,  24,  48,  is  an  ascending  geometrical  progression, 
because  48  is  as  many  times  greater  than  24,  as  24  is  greater 
than  12,  <Sbc. ;  and  250,  50,  10,  2,  is  a  descending  geometrical  pro- 
gression, because  2  is  as  many  times  less  than  10,  as  10  is  less  than 
50,  &c. 

In  the  first  mentioned  series,  (3,  6,  12,  24,  48,)  48  is  the  greater 
extreme,  and  3  is  the  less  extreme ;  the  numbers  6,  12,  24  are  tho 
means  in  that  progression. 

So,  too,  of  the  progression  250,  50,  10,  2 ;  250  and  2  are  the  ex- 
tremes, and  50  and  10  are  the  means. 

In  the  first  mentioned  progression,  2  is  the  ratio,  and  in  the  last, 
or  in  the  progression   2,  10,  50,  250,  5  is  the  ratio. 

In  a  geometrical  progression,  the  product  of  the  two  extremes  is 
equal  to  tho  product  of  any  two  means  that  are  equally  distant  from 
the  extremes,  and,  also,  equal  to  the  square  of  the  middle  term, 
when  the  progression  consists  of  an  odd  number  of  t<  ; 

Thus,  in  tho  progression  2,  6,  18,  54,  102  ;  102  X  2  =  5 1  X  r 
=  18  X  18. 

When  a  geometrical  progression  has  but  3  terms,  cither  of  the 


GEOMETRICAL   PROGRESSION.  151 

extremes  is  called  a  third  proportional  to  the  other  two ;  and  the 
middle  term,  consequently,  is  a  mean  proportional  between  them. 

Thus,  in  the  progression  48,  12,  3,  3  is  a  third  proportional  to  48 
and  12,  because  48  divided  by  the  ratio  =  12,  and  12  divided  by  the 
ratio  =  3  ;  or  3  X  ratio  =  12,  and  12  X  ratio  =  48  :  12  is  the  mean 
proportional,  because  12  X  12  =  48  X  3. 

Of  the  5  properties  of  a  geometrical  progression,  viz.,  the  greater 
extreme,  the  less  extreme,  the  number  of  terms,  the  ratio,  and  the  sum 
of  the  terms,  any  three  being  given,  the  other  two  may  be  found. 

Let  s  represent  the  sum  of  the  terms. 

E        "        the  greater  extreme. 

e         "        the  less  extreme. 

r         "         the  ratio. 

n         "         the  number  of  terms. 

n  when  affixed  as  an  index  or  exponent,  represent  that  the 
term,  number,  or  quantity,  to  which  it  is  affixed,  is  to  be  raised 
to  a  power  equal  to  the  number  of  terms  in  the  respective  progres- 
sion, &c. 

Any  three  of  tliefive  parts  of  a  geometrical  progression  being  given,  to 
find  the  remaining  two  parts. 

E  —  e 
-,  -(-Eat  sum  of  the  terms. 

— *  r      e  —  sum  of  the  terms, 
r— 1 

rn  X  c  —  e 


r— 1 


sum  of  the  terms. 


-\-  E  =  sum  of  the  terms. 
E  —  e 


^(E-he)-! 


■f-  E  =  sum  of  the  terms. 


Example.  —  The  greater  extreme  of  a  geometrical  progression  is 
162,  the  less  extreme  is  2,  and  there  are  5  terms  in  the  progression ; 
required  the  sum  of  the  series. 

80  +  162  =  242.    Ans. 


^(162-s-2)-r 


sXr-l+e 

=  greater  extreme. 


152  GEOMETRICAL  PROGRESSION. 

—  X  c  =  greater  extreme, 
r  n_l  X  c  =  greater  extreme. 


=  greater  extreme. 


r:i 


5  —  (5  —  E)  X  t  =  less  extreme. 
E  -f-  rn_1    =  less  extreme. 


sXr^Xr-L    .     n-1 

-£-r      =  less  extreme. 


r°— 1 
*  —  e 


s  —  E 


ratio. 


E 
n7j/-  =  ratio. 


*Xr 


1  =  rn ;  n,  therefore,  is  equal  to  the  number  of  timei 


sXr  —  l 

that  r  must  be  multiplied  into  itself  to  equal -f-  L 


-x—1     +1.,.. 


s—  (5  —  E)Xr 

Example.  —  A  farmer  proposed  to  a  drover  that  he  would  sell 
him  12  sheep  and  allow  him  to  select  them  from  his  flock,  provided 
the  drover  would  pay  1  cent  for  the  first  selected,  3  cents  for  the 
-second,  9  cents  for  the  third,  and  so  on  ;  what  sum  of  money  would 
12  sheep  amount  to,  at  that  rate? 

rn  X  e  —  e 

r  _l —  =  s}  then 

312x1  —  1 

jj^Tj— =  $2657.20.     Arts. 

Note.  —  Ratio4  ,  cubed  =  ratio12  ;  ratio6 ,  squared  =  ratio' ': ,  &c. 

When  it  is  required  to  find  a  high  power  of  a  ratio,  it  is  conven- 
ient to  proceed  as  follows,  viz. :  write  down  a  few  of  the  lower  or 
leading  powers  of  the  ratio,  successively  as  they  arise,  in  a  line,  one 
after  another,  and  placo  their  respective  indices  over  them  ;  then 


I J 


GEOMETRICAL  PROGRESSION.  153 

will  the  product  of  such  of  those  powers  as  stand  under  such  indices 
whose  sum  is  equal  to  the  index  of  the  required  power,  equal  the 
power  required. 

Example.  —  Required  the  11th  power  of  3. 

12     3      4       5 
3    9    27    81    243 

Here  5  +  4-f-2  =  ll,  consequently, 

243  X  81  X  9  =  11th  power  of  3,  or 

5X24-1  =  U»  consequently, 
243  X  243  X  3  =  11th  power  of  3,  or 

4X2  +  3  =  11,  consequently, 
81  X  81  X  27  =  11th  power  of  3,  or 

3  X  3-f-2  =  ll,  consequently, 

273  X  9  =  r11  =  177147.    Ans. 

To  find  any  assigned  number  of  geometrical  means,  between  two  given 
numbers  or  extremes. 

Rule.  —  Divide  the  greater  given  number  by  the  less,  and  from 
the  quotient  extract  that  root  whose  index  is  1  more  than  the 
number  of  means  required  ;  that  is,  if  1  mean  be  required,  extract 
the  square  root ;  if  two,  the  cube  root,  &c,  and  the  root  will  be  the 
common  ratio  of  all  the  terms  ;  which,  multiplied  bv  the  less  given 
extreme,  will  give  the  least  mean ;  and  that,  multiplied  by  the  said 
root,  will  give  the  next  greater  mean,  and  so  on,  for  all  the  means 
required.  Or  the  greater  extreme  may  be  divided  by  the  common" 
ratio,  for  the  greatest  mean ;  that  by  the  same  ratio,  for  the  next 
less,  and  so  on. 

Example.  —  Required  to  find  5  geometrical  means  between  the 
numbers  3  and  2187. 

2187  -7-  3  =  729,  and  ^729  —  3,  then  — 
3X3  =  9X3  =  27X3=81X3  =  243  X  3-=  729,  that  is,  the 
numbers  9,  27,  81,  243,  729  are  the  5  geometrical  means  between 
3  and  2187. 

Notb. —  The  geometrical  mean  between  any  two  given  numbers  is  equal  to  the  square 
root  of  the  product  of  those  numbers.    Thus  the  geometrical  mean  between  5  and  20,  =» 

V(5X20)=io. 


154  ANNUITIES. 


ANNUITIES. 

An  annuity,  strictly  speaking  and  practically,  is  a  certain  sum 
of  money  by  the  year ;  payable,  usually,  either  in  a  single  pay- 
ment yearly,  or  in  half,  half-yearly,  quarter,  quarter-yearly,  &c, 
and  for  a  succession  of  years,  greater  or  less,  or  forever.  Pensions, 
awards,  bequests,  and  the  like,  that  are  made  payable  in  fixed 
sums  for  a  succession  of  payments,  are  commonly  rated  by  the 
year,  and  denominated  annuities. 

A  current  annuity  that  has  already  commenced,  or  that  is  to 
commence  after  an  interval  of  time  not  greater  than  that  between 
the  stipulated  payments,  is  said  to  be  in  possession. 

One  that  is  to  commence  or  cease  on  the  occurrence  of  an 
indeterminate  event,  as  upon  the  death  of  an  individual,  is  a  re- 
versionary, contingent,  or  life  annuity. 

One  that  is  to  commence  at  a  given  period,  and  to  continue  for  a 
given  number  of  years  or  payments,  is  a  certain  annuity. 

One  that  is  to  continue  from  a  given  time,  forever,  is  a  perpetual 
annuity,  or  &  perpetuity. 

Annuity  payments  do  not  exist  fractionally :  they  mature,  and 
exist  only  in  that  state,  and  are  then  due. 

A  current  annuity  commences  with  a  payment,  and  terminates 
with  a  payment. 

One  current  in  the  past  is  measured  from  a  present  included  pay- 
ment, closes  with  an  included  payment,  and  is  said  to  be  in  arrears 
or  forborne,  from  a  supposed  cancelled  payment  one  regular  inter- 
val or  time  beyond. 

One  current  in  the  future  is  measured  from  the  present  to  the 
first  included  payment  of  the  series,  and  from  thence  is  said  to  con- 
tinue to  the  close ;  but  if  the  interval  from  the  present  to  the  first 
included  payment  is  equal  to  that  between  the  successive  pay- 
ments, it  is  supposed  to  continue  from  the  present. 

Annuities  in  negotiation  are  adjusted,  with  regard  to  time,  by 
interest,  or  discount,  or  both. 

The  tables  applicable  to  compound  interest  and  compound  dis- 
count are  applicable  in  adjusting  annuities  at  compound  rates. 

To  find  the  Amount  of  a  Current  Annuity  in  Arrears. 

Lemma.  —  The  amount  of  an  annuity  that  has  been  forborne 
for  a  given  time  is  equal  to  the  sum  of  the  several  payments  that 
have  become  due  in  that  time,  plus  the  interest  on  each,  from  the 
time  it  became  due,  until  the  close  of  the  time. 


ANNUITIES.  155 

Then  the  amount  of  an  annuity  of  $100,  payable  in  a  single 
payment  annually,  but  delayed  of  payment  4  years,  allowing  sim- 
ple interest  at  6  per  cent,  on  the  payments,  is 

100  X  1.18=  118 
100  X  1.12=  112 

100  X  1.06=  106 

100  X  1       =  100  =  $436. 

And  at  6  per  cent,  compound  interest  on  the  payments,  it  is 

100  X  (1.06)8  =  119.10 
100  X  (1.06)2  =  112.36 
100  X  (1.06)!  =  106.00 
100  X    1  =100.00  =$437.46. 

At  6  per  cent,  simple  interest,  when  payable  in  half,  half- 
yearly,  it  is 

50  X  1.21=60.50 
50  X  1.18  =  59.00 
50  X  1-15  =  57.50 
50  X  1.12  =  56.00 
50  X  1.09  =  54.50 
50  X  1.06  =  53.00 
50  X  1.03  =  51.50 
50  X  1       =  50.00  =  $442. 

And  at  6  per  cent,  compound  interest  per  annum,  when  payable 
in  half-yearly  instalments,  it  is 

50  X  (1.06)8  X  1.03=61.34 

50  X  (1.06)8  =59.55 

50  X  (1.06)2  X  1.03=57.86 

50  X  (1.06)2  =56.18 
50  X  (1.06)1  x  1.03  =  54.59 

50  X    1-06  =53.00 

50  X     1-03  =51.50 

50  X    1.  =50.00  =  $444.02. 

From  the  foregoing,  we  derive  the  following  general  Rules:  — 

Let  P  =  annuity  or  yearly  sum, 

r  =  rate  of  interest  per  annum, 

a  =  rate  of  discount  per  annum, 

n  or  n  =  nominal  time  of  the  annuity  in  full  years, 

A  =  amount  for  the  full  years, 

D  =present  worth  for  the  full  years. 


156  ANNUITIES. 

When  the  annuity  is  payable  in  a  single  payment  yearly, 
A  =  P/i  (l  -j-  ^f^)y  Simple  Interest 
A  =  P —     ~  i  Compound  Interest 

When  payable  in  equal  half-yearly  instalments, 

A  =  Pn  ( 1  -f  r-^~^  +  "f  )'  Simple  Interest. 

A  =  P  X  (1+ri.'"1X  ( 1  +  0'  Compound  Interest 
When  payable  in  equal  third-yearly  instalments, 

A  =  Pn  ( 1  +  ^Hz±  -j-  -f  ),  Simple  Interest 

A  =  P — £ —  (i_j_r^  Compound  Interest. 
When  payable  in  quarter-yearly  instalments, 

A  =  Pn  ( 1  +  r-^f^+  ~  ),  Simple  Interest. 

A  =  P  (1+r^~1  ( l  -f  ^- ),  Compound  Interest. 
When  there  are  odd  payments,  to  find  the  amount,  S. 


When  1  half-yearly,      S  ==  A(l 


1  third-yearly,      S  =  A(l  +  £  r) 


*')  +  *?• 


S  =  A(l--}r)--JP. 

S  =  A(l-|-fr)-JP(l  +  W+JP 

,S  =  A(l-^r)-iP.    ,     ^ 


±r)  +  P(8  +  r)-^16. 


1  quarter-yearly, 

2  ".  S  =  A(1 

3  «  S  =  A(l  +  |r)  +  P(3  +  |r)-f-4. 

For  any  number  of  equal  and  regular  payments  at  compound 
interest  per  interval  between  the  payments,  S  =  P'  ( — - — j,  and 
for  any  number  of  equal  and  regular  payments  at  simple  interest 
per  interval  between  the  payments,  S  =  P'n'  (l  -\-  r'lH'~1))  ;  P' 
being  a  payment,  n1  or  n'  the  number  of  payments,  and  r1  the  rate 
of  interest  per  interval  between  the  payments.  But  this  must  not 
be  confounded  with  compound  interest  annually,  on  payments  oc- 
curring semi-annually,  quarterly,  &c. 

Example.  —  What  is  the  amount  of  an  annuity  of  $150,  paya- 
ble in  half,  half-yearly,  but  delayed  of  payment  2  years  and  72 
days,  allowing  compound  interest  per  annum  at  7  per  cent.  V 

150  xl^~=  $310.50,  the  amount  for  2  years,  if  payable 
in  yearly  payments,  and 


ANNUITIES.  157 

810.50X  0-^)  =$315.93,  the  amount  for  2  years,  if  payable 
in  half-yearly  payments,  and 

815.93  X  O'^^O  =$320.29,  the  amount  for  2  years  and  72 
days,  if  payable  in  half-yearly  payments.     Ans. 

Example.  —  What  is  the  amount  of  an  allowance,  pension,  or 
award,  of  $100  a  year,  payable  quarterly,  but  forborne  3£  years, 
interest  compound  per  annum  at  6  per  cent.  ? 

100  X  (-^—  X  (  1  -f1^)  =  $325.52*,  the  amount  for  3  years, 

and 

325.52  (1  -{-  .03)  -J-100X  8.06  -f- 16=  $385.66.     Ans. 

Example.  —  What  is  the  amount  of  $100  a  year,  payable  in 
quarterly  payments,  and  in  arrears  4  years,  interest  being  com- 
pound per  quarter-year,  at  6  per  cent,  a  year  ? 

25  [(1  -f-f)1- 1]  X  ~m  •    Bv taDular  powers  of  (1  +  r),  page 

125,  =$448.30.     Ans. 

To  find  the  Present  Worth  of  an  Annuity  Current. 

Lemma.  —  The  present  worth  of  an  annuity  that  is  to  continue 
for  a  given  time  is  equal  to  that  sum  of  money,  which,  if  put  at 
interest  from  the  present  time  to  the  close  of  the  payments,  will 
amount  to  the  amount  of  the  payments  at  that  time ;  and  therefore, 
the  times  being  full,  is  equal  to  the  sum  of  the  several  payments, 
discounted,  respectively,  at  the  rate  of  interest  for  their  respective 
times. 

Note.  —If  the  foregoing  proposition  is  tenable,  it  follows,  since  simple 
interest  is  due  and  payable  annually,  that  the  true  present  worth  of  an  annuity 
having  more  than  one  year  to  run  cannot  be  found  by  simple  interest  and  dis- 
count. By  simple  interest  and  discount,  at  6  per  cent.,  predicating  the  rule 
upon  the  foregoing  lemma,  the  amount  of  $100,  payable  annually,  and  in 
arrears  for  4  years,  is  $436  5  and  the  present  worth,  at  6  per  cent.,  is 

jpo    ioo_    wo     100 

1.24^1.18      1.12^1.06       * 

But  $349  at  6  per  cent,  interest  for  4  years,  with  the  payments  of  interest  annu- 
ally, will  amount  to  $440.60;  and  at  interest  simply  for  4  years  it  will  amount 
to  only  $432.76. 

Then  the  present  worth  of  an  annuity  of  $100,  payable  in  a 
single  payment  yearly,  and  to  continue  4  years,  or  to  become  due 
1,  2,  3,  and  4  years  hence,  interest  and  discount  being  compound 
per  annum,  and  each  at  6  per  cent.  = 
14 


158  ANNUITIES. 

P  P  P  P 

U  +  /T-L-TT3  +  n-j-^2  +  i-T- =  S346.51  =z 


(1+r)*  '    (l+r)s  '    (i+r)»^l+r- 

100  X  (1.06)3  =  119.10 

100  X  (1.06)2=  112.36 

100  X  (1.06)  =106.00 

100  X    1  =  100.00  =  437.46 -t-(1.06)4z=  $346.51. 

And  interest  at  6  per  cent,  and  discount  at  10,  both  compound,  it  is 
100  X  (1.06)3=  119.10 
100  X  (1.06)2=  112.36 
100  X    1-06     =106.00 
100  X    1  =100.00  =  437.46  -f-(1.10)4=  $298.79. 

Therefore,  when  the  annuity  is  payable  in   a  single  payment 
yearly  from  the  present  time, 

D  =  P (  "^r)~1»=  n  when  r  and  a  are  equal. 

ra  +  af        a  +  rr  ^ 

When  payable  in  half-yearly  payments, 

D=Px<wx(1  +  ir>- 

When  payable  in  third-yearly  payments, 

-p  __ PxCd+D'-llxd-Hr^ 
r(l  +  a)s 
When  payable  in  quarter-yearly  payments, 

D  =  rftl  +  r)*-l](l  +  jr). 
r(l  +  a)n 

When  there  are  odd  payments,  to  find  the  present  worth,  S. 
There  being  a  half-yearly,  S  =  l-^r-  -f-  ,-4^— 
«         1  third-yearly,  8  =  ,-^-+^. 

«  9  «  S J—    I  2P(l+$r). 

o-_1  +  |a-r  8(1  +  fa) 

«  1  quarter-yearly,  S  =  q^+  r^Ta 

Ml  "  S  —  -5-  _L  Z&±i!$* 

For  any  number  of  equal  payments,  at  equal  intervals  between 
the  payments,  S  =  P'X(~^ri   P'  being  a  payment,  n'  the 


ANNUITIES.  159 

number  of  payments,  and  r'  and  a'  the  rates  per  interval  between 
the  payments. 

Note.— Since  (1~*"r)   71  is  the  co-efficient  of  V,  for  its  present  worth,  at 

compound  interest  and  discount,  for  the  time  »  ,  at  the  rates  r,  «,  it  follows 
that  tables  of  co-efficients  of  1*  lor  its  present  worth,  at  given  rates,  for  any 
number  of  years,  may  be  easily  made.  Thus  (1-06*  —  n-s-i.oe4  x.06  =  :u(;r>n, 
the  co-efficient  of  an  annuity,  P,  for  4  years'  continuance,  interest  and  discount 
being  compound  per  annum,  at  6  percent.;  and  (1.062  —  D-Ml.062  X  -06)  = 
1.83339,  the  co-efficient  for  2  years,  &c. 

If  the  annuity  is  deferred,  then  the  diflerenceof  two  of  these  co-efficients 
(one  of  them  that  for  the  time  deferred,  and  the  other  that  for  the  sum  of  the 
time  deferred  and  the  time  of  the  annuity)  will  be  the  co-efficient  of  P  for 
its  present  worth.  Thus  3.4051 1  —  1.83339*=  1.63172,  the  co-efficient  of  an  annuity, 
P,  for  its  present  worth,  when  it  is  to  commence  two  years  hence,  and  to  con- 
tinue 2  years,  interest  and  discount  being  compound  per  annum,  at  G  per 
cent,  each;  or  D=  1.03172  P. 

In  like  manner,  tables  of  other  co-efficients,  such  as  the  formulae  suggest, 
may  be  made  that  will  greatly  assist  in  calculating  annuities. 

Example.  —  What  is  the  present  worth  of  an  award  of  $500  a 
year,  payable  in  half-yearly  instalments,  the  1st  payment  to  mature 
6  months  hence,  and  the  annuity  to  continue  three  years;  interest 
and  discount  being  7  per  cent.,  compounded  yearly  ? 

500  X  [(1.07)3— 1]X  (l.f) 

>oyx(1>07),-^—^  =$1888.18.    Ans. 

Example.  —  What  is  the  present  worth  of  an  annuity  of  $100, 
payable  in  half-yearly  payments,  and  to  continue  1£  years;  interest 
and  discount  being  6  per  cent,  per  annum  ? 

100  X  [1.06  —  l]Xl-46 

D  — - 95  755   ana 

U  —  .06X1.06  XW,7W' 

95.755   ,     50 

Tor+roa^*141-61-  Ans- 

Example.  —  What  is  the  present  worth  of  an  annuity  of  $500, 
payable  in  semi-annual  instalments,  and  to  continue  10|  years, 
interest  and  discount  being  compound  per  annum,  the  former  at  6 
per  cent.,  and  the  latter  at  8  ? 

500  [(1.06)*- 1]  (I..*)  5QQ 

.06(1.08y(l.f)         +2(l.f) 

A  250 

1.0810  X  1-04  +  1.04  —     AnS' 


160  ANNUITIES. 

KDwers  of  1  -}-  r,  page  12 
=  $3052.64,  the  present  worth  for  10  years'  con- 


By  tabular  powers  of  1  -}-  r,  page  125 : 
500  X  -79085 


.06  X  2.15892 

tinuance,  if  payable  in  yearly  payments,  and 

3052.64  X  1-015  =  $3098.43, 

the  present  worth   for  10  years'  continuance,  if  payable  in  half- 
yearly  payments,  and 

3098.43  -7-  1.04  -|-  500  -+-  2  X  1-04  =  $321 9.64.     Ans. 

When  the  interval  of  time  from  the  present  to  the  1st  payment 
is  shorter  than  that  between  the  consecutive  payments,  and  the 
annuity  is  payable  in  a  single  payment  yearly, 

A=P[(l+r)--l](l+£) 

r 

A  P  [(!  +  »•)■- !](!+£) 

d  being  the  time  in  days  from  the  present  to  the  1st  payment. 

So,  if  the  annuity  is  payable  in  half-yearly,  third-yearly,  or  quar- 
ter-yearly instalments,  multiply  by  1  4-  £  r,  1  -J-  ^  r,  or  1  -}-  f  r,  as 
before  directed ;  and  if  there  are  odd  payments  proceed  for  the 
present  worth,  S,  as  already  directed. 

Example.  —  Required  the  present  worth  of  an  annuity  of  $100, 
payable  yearly,  to  commence  4  months  hence,  and  to  continue  4 
years ;  interest  and  discount  being  6  per  cent  annually. 

100  X  (1.064  —  1)  X  (l-1^) 

.06  X  1-063  x  (i.--2^r^) 

To  find  the  Present  Worth  of  a  Deferred  Current  Annuity ■,  or  of 
an  Annuity  in  Reversion. 

When  the  annuity  is  payable  in  a  single  payment  yearly,  and  the 
deferred  time  embraces  full  years  only, 

D  =  P  i1  ~r  r)   ""         n'  being  the  deferred  time. 
r(l  +  o)<B+B) 

If  it  is  payable  in  half-yearly,  third-yearly,  or  quarter-yearly 
instalments,  multiply  by  1  -f  \  r,  1  +  |  r,  or  1  +  §  r,  as  already 


ANNUITIES.  161 

directed ;  and,  if  there  are  odd  payments,  find  the  present  worth,  S, 
as  already  directed. 

Example.  —  What  is  the  present  worth  of  an  annuity  of  $150, 
payable  yearly,  to  commence  2  years  hence,  and  to  continue  4 
years ;  interest  and  discount  being  compound  per  annum,  at  6  per 
cent.  ? 

150  X  (1-064  —  1)  -r  .06  X  1-066  =  $462.59.     Am. 

Example.  —  Required  the  present  worth  of  an  annuity  of  $500, 
payable  in  semi-annual  instalments,  to  commence  2£  years  hence, 
and  to  continue  6  years  ;  allowing  compound  interest  and  discount 
annually  at  7  per  cent. 


500  X  (1.078—  1)  X  l.,Jj- 
.07  X  1.078  X  1. '°' 


$2046.44.     Ans. 


2 

Example.  —  Required  the  present  worth  of  an  allowance, 
pension,  or  award  of  $125  a  year,  payable  in  half  every  half-year, 
to  commence  7  months  24  days  hence,  and  to  continue  6£  years  ; 
interest  and  discount  being  compound  per  annum  at  5  per  cent. 

125  X  (1.05°  -1)X  1.0125  125 

.05  X  1-056  X  1-03247  X  1025  ~  2  X  1-025 

0r  ^^^xaow^  =  S634,47'  the  Presenfc  worth  for  6 
years'  continuance,  if  payable  in  yearly  instalments  ;  and 

634.47  X  l-x  =  &642.40, 

the  present  worth  for  6  years'  continuance,  if  payable  in  half-yearly 
instalments  ;  and 

642.40^(1  +  ^^?)  =  622.20, 

the  present  worth  for  6  years'  continuance,  if  payable  in  half-yearly 
instalments,  and  to  commence  7  months,  24  days  hence ;  and 

622.20  -^(l+-5i)+2_^_  =  $668.     Ans. 


To  jind  the  Present  Worth  of  a  Perpetuity. 

Lemma.  —  The  present  worth*  of  an  annuity  to  commence  one 
year  hence,  and  to  continue  forever,  is  expressed  by  that  sum  of 
money  whose  interest  for   1  year  is  equal  to  the  amount  of  the 
14* 


*62  annuities. 

annuity  for  1  year ;  and  so,  pro  rata,  for  perpetuities  otherwise 

regularly  affected. 

Then  when  the  annuity  is  to  commence  1  year  hence,  and  is 

payable  in  a  single  payment  yearly     .     .     .     D  =  P  -J-  r. 

P(l  -4-  ±r) 
Payable  in  half-yearly  instalments       .     .    D  =        ~        . 


Payable  in  third-yearly  instalments 


Pfl  +  fr) 


PCl  4-  #r) 
Payable  in  quarter-yearly  instalments      .     D  =    v     '   g    ■ 

r 

Example.  —  What  is  the  present  worth  of  a  perpetuity  of  $150 
a  year,  payable  in  a  single  payment  yearly  from  the  present  time  ; 
interest  at  6  per  cent  ? 

150  -f-.06  =$2500.     Ans. 

Example.  —  What  is  the  present  worth  of  a  perpetuity,  of  $150 
a  year,  payable  in  semi-annual  instalments,  and  to  commence  4 
months  hence  ;  interest  7  per  cent  ? 

PO+iO- 07(12^  =3(,      Am_ 
r  '  12 

Example.  —  Required  the  present  worth  of  a  perpetuity  of 
$400  a  year,  payable  in  quarterly  payments,  and  to  commence  6 
years  hence ;  interest  and  discount  being  5  per  cent.,  compound 
per  year. 

P(l-J-fr)       400  Xl.3-^ 

D =  — — £-r  =  $6081.65.     Ans. 

u  —  r(l-f-a)B  .05X1-058 


The  Amount,  Time,  and  Rate  given,  to  find  the  Annuity. 
When  payable  in  a  single  payment  yearly  from  the  present  time, 

* = (i+^=r « hM-y^' p  -  Ki+ry-i]  <!+*■) ; 

third-yearly,  P  =  y+^+^q  i  quarterly, 

P  = 


(l  +  if<-)[(l  +  r)--l]  ' 


ANNUITIES.  163 

and  so,  pro  rata,  for  other  fractional  units  of  the  integral  unit. 

™.  x  Ar  Ar  Ar 

Therefore  (l  +  r).-l=ir,   „__,OTf-_ 

orpTTF)'&c- 

Example.  —  What  annuity,  payable  in  quarterly  payments 
from  the  present  time,  will  amount  to  $3000  in  12  years;  interest, 
being  compound  per  annum,  at  8  per  cent.  ? 

3000  X  -08  -|-  [(10812  —  1)  X 1. 3-JV£\l  =  $153-48-     Ans- 

Example.  —  What  length  of  time  must  a  current  annuity  of 
$400,  payable  in  quarterly  payments,  remain  unpaid,  that  it  may 
amount  to  $2500 ;  interest  being  7  per  cent,  yearly  ? 

2500 X  .07  =  ,4263094  =  5  4-  years,  and  5  years  by  table  of 
400  XL ^x27 

(l+*-l-  .402552  :  therefore  fgjg-  1)  3J£=  308 

days,  5  years,  308  days.     Ans. 


The  Present  Worth,  Time,  and  Rate  given,  to  find  the  Annuity. 

When  payable  in  a  single  payment  yearly  from  the  present  time, 
_      Dr(l+r)n    .'  .     „  Dr(l4-r)n  .... 

*~0+ JL-1    »  half^  P  -  T(l+r)^l](l+ir)  5  third" 

yearly,         P  =  [(l  ^^+^+  ,r)  5  quarter-yearly,  P  = 

[(1+y^()l  +  IO-&C-     Therefore,  (l+^=p^  = 
P(l+jr)        _         P(l  +  |r) 
P(l_L.ir)_Dr~~P(l  +  fr)  — Dr  '      ' 

Example.  —  What  annuity,  payable  in  half-yearly  instalments, 
and  to  continue  3  years,  is  at  present  worth  $1335.13  ;  discount  and 
interest  being  compound  per  year,  at  7  per  cent  ? 

1335.13X.07X1-073        ..^        . 

— w  =  $500.     Ans. 


(1.073— 1)X1-^ 


164  ANNUITIES. 


OF   INSTALMENTS   GENERALLY. 

Any  certain  sum  of  money  to  be  paid  on  a  debt  periodically 
until  the  debt  is  paid  is  called  an  instalment ;  and  a  debt  so  made 
payable  is  said  to  be  payable  by  instalments. 

Let  D  =  principal  or  debt  to  be  paid, 

n  =  number  of  years  in  which  the  debt  is  to  be  paid, 

r  a=  rate  of  interest  per  annum, 

p  =  instalment  or  periodical  payment. 

When  the  instalments  are  payable  yearly,  and  the  debt  u  at 
interest, 

When  payable  half-yearly, 

Dr(l-fr)" 


2[(l+r)--l](l  +  ir]1 


1 f*  "[^(1+ W+rf  -Dr '  "  -  r(l  +  r)« 

When  the  debt  is  not  on  interest,  and  the  instalments  are  pay- 
able yearly, 

f-(i  +  f)..i'(!  +  r)  =-7-'D- F * 

Example.  —  What  yearly  instalment  will  pay  a  debt  of  $4000 
in  4  years,  the  debt  being  on  interest  the  while,  at  6  per  cent, 
annually  ? 

4000  X  -06  X  1.064  -7-  (1.064  —  1)  =$1154.37.     Ans. 

Example.  —  What  semi-annual  instalment  will  pay  a  debt  of 
$4500  in  3  years,  the  debt  bearing  interest  at  7  per  cent,  yearly  ? 

4500  X  .07  X  1.07'       _  $842.62.     Am. 


2  X  (1-07*  —  1)X  1-0175 


ANNUITIES.  165 

When  a  debt  has  been  diminished  at  regular  intervals  by  the 
payment  of  a  constant  sum,  to  find  the  remaining  debt  at  the  close 
of  the  last  payment. 

When  the  debt  is  on  interest,  and  the  payments  have  been  made 
yearly  from  the  date  of  the  debt, 

P—      (i_j_r)»_i     ' 

When  the  payments  have  been  made  half-yearly, 

d=p+p(l  +  $r)-(l  +  ry[p  +  p(l+ir)-Dr]  +  r,lkc. 

Example.  —  On  a  debt  of  $1000,  drawing  interest  the  while 
at  8  per  cent,  a  year,  there  has  been  paid  yearly,  from  the  date  of 
the  debt,  $200  for  6  years :  required  the  unpaid  debt  at  the  close 
of  the  last  payment. 

[200—  1.086(200  —  .08  X  1000)]  +  .08  =  $119.69.     Ans. 

Example.  —  On  a  note  of  hand  for  $1000,  and  interest  from 
date,  at  8  per  cent,  annually,  the  following  payments  have  been 
made;  viz.,  $100  at  the  close  of  every  half-year  from  the  date  of 
the  note,  for  6  years.  How  much  remained  unpaid  at  the  close 
of  the  last  payment  ? 

[204  —  1.088(204  —  .08  X  1000)]  -f-  .08  =  $90.34.     Ans. 


166  PERMUTATION. 


PERMUTATION. 

Permutation,  in  the  mathematics,  has  reference  to  the  greatest 
number  of  unlike  relative  positions,  that  a  given  number  of  things, 
either  wholly  unlike,  or  unlike  only  in  part;  may  be  placed  in.  It 
considers  the  number  of  changes,  therefore,  that  may  be  made,  in 
the  arrangement  of  the  things,  under  different  given  circumstances. 

To  find  the  number  of  changes  that  can  be  made  in  the  order  of  arrange- 
ment of  a  given  number  of  things,  when  the  things  are  all  different. 

Rule.  —  Find  the  product  of  the  natural  series  of  numbers,  from 
1  up  to  the  given  number  of  things,  inclusive  ;  and  that  product  will 
be  the  number  of  changes   or  permutations   that  may  be  made. 

Example.  —  In  how  many  different  relative  positions  may  12  per- 
sons be  seated  at  a  table  ? 

1X2X3X4X5X6X7X8X9X10X11X12  = 

479,001,000.     Am. 

To  find  the  number  of  changes  that  can  be  made  in  the  order  of  arrange- 
ment of  a  given  num/jer  of  things,  when  that  number  is  composed  of 
several  different  things,  and  of  several  which  are  alike. 

Rule.  —  Find  the  number  of  changes  that  could  be  made  if  the 
things  were  all  unlike,  as  in  first  example.  Then  find  the  number 
of  changes  that  could  be  made  with  the  several  things  of  each  kind, 
if  they  were  unlike.  Lastly,  divide  the  number  first  found  by  the 
product  of  the  numbers  last  found,  and  the  quotient  will  be  the 
number  of  permutations  or  changes  that  the  collection  admits  of. 

I  a  ample.  —  Required  the  number  of  permutations  that  can  be 
made  with  the  letters  a,  bb,  ccc,  dddd,  =  10  letters. 

1X2X3X4X5X6X7X8X9X10  =  3028800 

1  X  2  X  6  X  24  =     288     -  uTn'  Ans' 

To  find  the  number  of  permutations  that  can  be  made  with  a  given  num- 
ber of  different  things,  by  taking  an  assigned  number  of  them  at 
a  time. 

Rule.  —  Take  a  series  of  numbers  beginning  with  the  number  of 
things  givrn,  :ind  decreasing  by  1  continually,  until  the  number  of 
t  this  is  equal  to  the  number  of  things  that  are  to  be  taken  :it  :i 
time  ;  then  will  the  product  of  the  scries  be  the  number  of  changes 
that  may  be  made. 


COMBINATION.  '  1()7 

Example.  —  What  number  of  changes  can  be  made  with  the 
numbers  1,  2,  3,  4,  5,  6,  taking  three  of  them  at  a  time? 

G  —  1  =  5,  5  —  1  =  4,  then  0X5X4  =  120.     Ans. 
What  number,  by  taking  4  of  them  at  a  time  1 
0X5x4X3  =  3C0.    Ans. 

Example. — Arrange  the  three  letters  a,  b,  c,  into  the  greatest 
number  of  permutations  possible. 

abc,  acb,  bac,  bca,  cab,  cba,  =  6  permutations.     Ans. 

Example.  —  Arrange  the  four  letters  a,  b,  a,  b,  into  the  greatest 
number  of  permutations  possible. 

abab,  aabbf  abba,  bbaa,  baba,  baab,  =  G  permutations.     Ans 


COMBINATION. 

Combination,  in  the  mathematics,  lias  reference  to  the  number  of 
unlike  groups,  which  may  be  formed  from  a  given  number  of  differ- 
ent things,  by  taking  any  assigned  number  of  them,  less  than  the 
whole  at  a  time.  It  does  not  regard  the  relative  positions  of  the 
things,  one  with  another,  in  any  of  the  collections  or  groups.  But 
it  exacts  that  each  group,  in  all  instances,  shall  have  the  assigned 
number  of  members  in  it,  and  that,  in  every  group,  in  every  instance, 
there  shall  be  a  like  number  of  members.  It  exacts,  therefore,  that 
no  two  groups  shall  be  composed  of  precisely  the  same  members. 

To  find  the  number  of  combinations  that  can  be  made  from  a  given 
number  of  different  things,  by  taking  any  given  number  of  them  at  a 
time. 

Rule. — Take  a  series  of  numbers  beginning  with  that  which  is 
equal  to  the  number  of  things  from  which  the  combinations  are  to 
be  made,  and  decreasing  by  1,  continually,  until  the  number  of 
tonus  is  equal  to  the  number  of  things  that  are  to  be  taken  at  a  time, 
and  find  the  product  of  those  numbers  or  terms.  Then  take  the  natural 
series,  1,  2,  3, 4,  &c,  up  to  the  number  of  things  that  are  to  be  taken 
at  a  time,  and  find  the  product  of  that  series.  Lastly,  divide  the 
product  first  found  by  the  product  last  found,  and  the  quotient  will 
express  the  number  of  combinations  that  can  be  made. 

Example.  —  What  number  of  combinations  can  be  made  from  8 
different  things,  by  taking  4  of  them  at  a  time? 
8  X  7  X  G  X  5  _  1680 
IX  2X3X4""    24    —  70-     Ans- 


56.    Ans. 


168  COMBINATION. 

What  number,  by  taking  5  of  them  at  a  time  % 

8X7X6X5X4      6720 

1X2X3X4X5"  120 
What  number,  by  taking  3  of  them  at  a  time  ? 

8  X  7  X  6  =  336 

1X2X3         6   -&b*     Ans' 

Example.  —  What  number  of  combinations  can  be  made  from  5 
different  things,  by  taking  three  of  them  at  a  time  ? 

5X4X3       60 
1  X  2  X  3  =  6""10*    Ans' 
What  number,  by  taking  2  of  them  at  a  time  ? 

A*i-?!UlO.     Ans. 
1X2       2 

Example.  —  Form  5  letters,  a,  b,  c,  d,  e,  into  10  combinations  of  2 
letters  each ;  that  is,  into  10  unlike  groups  of  two  letters  each. 

ab,  aCj  ad,  ae,  be,  bd,  be,  cd,  ce,  de.     Ans. 

Form  them  into  the  greatest  number  of  combinations  possible,  in 
collections  of  three  each. 

abc,  obi,  abe,  acd,  ace,  ode,  bcdf  bee,  bde,  cde.     Ans. 


FOREIGN  MONEYS   OF  ACCOUNT: 

THEIR     DENOMINATIONS,    RELATIVE   VALUES,    AND    VALUES     IN     FEDERAL 

MONEY. 

The  value  in  Federal  money,  affixed  to  any  particular  denomina- 
tion of  a  Foreign  money  of  account,  in  the  following  tables,  is  the 
intrinsic  par  thereof,  as  near  as  practicable.  It  is  based  upon  the 
standard  weight  and  purity  of  the  coins  coined  especially  to  repre- 
sent that  denomination,  compared  with  the  standard  weight  and 
purity  of  the  coins  of  the  United  States  that  represent  the  dollar, 
and  is  the  United  States  Customs  value  of  that  denomination  for 
computing  duties.  The  denomination  itself,  to  which  the  Federal 
value  is  immediately  affixed,  is  usually  the  unit  or  ultimate  money 
of  account  of  the  country  especially  referred  to.  It  is  a  money  of 
account  in  that  country  always ;  but  not  always  in  that  country 
the  name  of  a  national  coin.  It  is  not  always,  even,  represented  by 
a  single  national  coin.  Thus,  in  Great  Britain,  until  comparatively 
recently,  there  was  no  single  British  coin  of  the  value  of  one  pound. 
That  denomination  is  now  represented  by  a  single  gold  coin,  called 
a  sovereign. 

Foreign.  U.  States. 

ALGIERS.  —  Algiers,  Bona,  &c. :  100  centimes  =  1  Li- 

vre, =  $0,187 

29  aspers  =  1  tomin,  8  tomins  =  1  pataka-chicas, 

3  patakas-chicas  =  1  Piastre,  -  -        =0.21 

ARABIA.  —Mocha,  Jidda :  80  caveers  =  1  Piastre,        -  =    0.823 

1  wakega/or  gold  and  silver  =  480  troy  grains. 
AUSTRIA.  — 100  centesimi  =  1  Lira  Austriache,   -        =    0.1622 

Vienna,  Trieste,  Prague  (Commercial)  :  4  pfennige 
=  1  kreuzer,  GO  k.  =  1  Gulden  or  Florin  Austri- 
ache =  j\y  of  a  Cologne  mark  of  fine  silver  = 
180-fVb-  troy  grains,         -  -  -  -  =    0.4858 

6  hellers  =  1  groschen,  80  g.  =  1  Florin,  -        =    0.4858 

1 J  groschen  =  1  kreuzer. 

l|  florins  =  1  current  Thaler. 

2  florins  =  1  Specie  Thaler. 

1  florin  Austriache  =  5fy  lire^de  Trieste  =  5yV  lire 

di  piazza. 
1  mark  Austrian  =  4332|  troy  grains. 


2  a  FOREIGN   MONEYS  OP   ACCOUNT. 

Foreign.  U.  States. 

AZORE  ISLANDS.  —  Corvo,  Fayal,  Flores,  Graciosa, 
Pico,  Terceira,  St.  Michael,  St.  Mary's:  1000 
reas  =  1  Milrea,  =  $0.83  J 

BALEARIC  ISLANDS.  —  Majorca  I.  —  Palma :  Mi- 
norca I.  —  Port  Mahon  :  same  as  Cadiz,  Spain. 

BELGIUM.  —  Antwerp,     Brussels,     Liege,     Mechlin, 
Ghent,  &c.  : 
10  centimes  =  1  decime,  10  d.  =  1  Franc,        -        ■■    0.187 
24  mitres  or  8  Brabant  penning  or  6  liarde  or  2 
groote  =  1  stuiver,  6  stuivers  =  1  scheiling,  20  s. 
=  1  Pond  Flemish,  -  -  -    '        -  =    2.377 

6  gulden  or  2£  daalder  =  1  Pond  Flemish. 

BERMUDAS  I.  —4  farthings  =  1  penny,  12  pence  =  1 

shilling,  20  shillings  =  1  Pound,       -  -        =    3.00 

BOURBON   ISLANDS.  —  100  centimes  =  1  Franc,     -  =    0.187 

BRAZIL.  —  Aracati,  Bahia,  Maranharn,  Para,  Pernam- 
buco,  Portalegra,  Rio  Janeiro,  &c.  :  1000  reas  = 
1  Milrea, =    1.042 

BONAIRE  I.  —  48  stuivers  or  8  reals  =  1  Piastre,        =    0.73 

Central  and  South  America. 

Balize,  Campeche,  Guatimala,  Honduras,  Laguna,  Leon, 

Nicaragua,  San  Juan,  San  Salvador,  Sisal,  &c.  : 
Buenos  Ay  res,  Callao,  Carthagena,  Coquimbo,  Guayaquil, 
Laguayra,    Lima,   Maracaybo,   Montevideo,    Rio 
Hacha,  Truxillo,  Valparaiso,  &c.  : 
2  cuartilli  =  1  medio,  2  medios  =  1  Real,        -        =    0.12^ 
8  reale  =  1  Peso,     -  -  -  -  -=     1.00 

Pound  of  Honduras,      -  -  -  -        =    3.00 

Berbice,  Demerara,  Essequibo,  Surinam  : 

8  duyt  =  1  stuiver,  20  s.  =  1  Florin  or  Gulden,       =    0.33J 
Cayenne:  100  centimes  =  1  Franc  or  Livre,  -  -  =    0.187 

CANARY     ISLANDS.  —  Grand     Canary,     Teneriffe, 
Palma,  &c.  : 
34  maraw.li  a  1  Real  (current),  -  -         =     0.074 

CAND1A   I.  —  80  aspers  or  33  niedini  =  1  Piastre,        =    0.048 
CAPE   VERD   I.  —  1000  reds  =  1  Milrea,        -  -  = 

CHILL — Coquimbo,   San    Carlos,    Santiago,    Valdiiia, 
Valparaiso,  &c.  : 
4  cuartilli  =  1  medio,  2m.  =  l  real,  8r.  =  l  Peso,  =     1.00 
CHINA.  —  Amoy,    Canton,    Macao,    Nankin,     Pckin, 
Shanghae,  &c.  : 
10  cash  =  1  <  amlarine,  10  candarines=  1  mace,  10 

bum*  =  1  Tael,  ----■■    1.48 

Hong  Kong  I.  —  Same  as  Great  Britain. 


FOREIGN   MONEYS  OF  ACCOUNT.  a  3 

Foreign.  U.  States. 

CORSICA  I.  —  100  centimes  «=  1  Franc,  -  -  =  $0,187 

CYPRESS  I.— Same  as  Constantinople  (Turkey). 
DENMARK.  —  Copenhagen,  &c.  :  12  pfennige  ==  1  skil- 
ling,  16  s.  =  1  mark,  G  m.  =  1  Ry  ksbankdaler  = 
■fir  of  a  Cologne  marc,  or  194.98  troy  grains  of 
pure  silver,  -  -  ■■  -  -  =    0.5252 

2  Ryksbankdalers  =  1  Speciesdaler,      -  -        =1.05 

EGYPT.  —Alexandria  and  the  Delta :  8  borbi  or  6  fiorli 

or  3  aspers  =*  1  medimne,  40  m.  =  1  Piastre,        =    0.048 
20  piastres  =  1  real  (a  gold  coin),  -  -  -  =     0.908 

Cairo :  80  aspers  or  33  medimni  =  1  Piastre,       -        =     0.048 
FRANCE.  —  Standard  for  gold  and  silver  coins  =  ^ 
fine,  each  :  Relative  values  —  gold  to  silver  as  15 
to  1. 
10  centimes  =»  1  decime,  10  d.  =*  1  Franc  =  stkt^ 

kilogrammes  of  fine  silver  =  G9.449  troy  grains,    =  0.18706 
12  deniers  =  1  sou,  20  s.  =  1  Livre  tournois  (old) 
=  |$  franc. 
GERMANY.  —The  mark  of  Cologne,  divided  into  8  unze  =  64 
quent  =  256  pfennig  «  512  heller  =  4352  eschen  =  65536  richt- 
pfennig,  is  employed  at  the  mints,  for  weighing  gold  and  silver 
coins,  throughout  Germany  :  this  mark  =  3607|  troy  grains. 

The  standard  for  the  Ducat,  at  present,  throughout  Germany,  is 
yj-  or  23|  carats  fine,  its  weight,  ^V  of  a  Cologne  mark.  This  piece, 
therefore,  should  weigh  53.84  troy  grains,  and  contain  53.09  troy 
grains  of  fine  gold  =  $2,286. 

The  standard  for  the  Pistole  or  Zehn  Gulden  piece,  of  the  Con- 
vention of  1753,  is  f  £  or  21|  carats  fine  ;  its  weight,  ^V  of  a  Cologne 
mark.  This  piece,  therefore,  should  weigh  103.06  troy  grains,  and 
contain  93.4  troy  grains  of  fine  gold  =  $4,022. 

The  standard  for  the  Specie  ThaJer,  or  Rixdollar  affective,  of  the 
Convention  of  1753,  is  |  or  13  loth  6  gran  fine,  its  weight  2\  Co- 
logne mark  =  $0.9716 ;  and  the  species  Florin  is  £  thereof  =* 
$0.4858. 

The  standard  value  of  the  Current  Thaler  or  Rixdollar  of  account, 
established  about  1775,  is  |  Rixdollar  affective  =  l£  specie  Florins 
=  $0.7287. 

The  Thaler  of  the  Convention  of  1838  contains  yj-  Cologne  mark 
of  fine  silver  =  $0,694  ;  and  the  Gulden  or  Florin  of  that  Conven- 
tion =  f  thereof  =  $0.3966  :  1|  gulden  =  1  Thaler. 

The  standard  of  purity  for  the  coins  established  by  the  Convention 
of  1838,  are  :  Two  Thaler  piece  =  T9^  ;  thaler,  f  thaler,  £  thaler, 
each,  =  | ;  florin,  £  florin,  each  =  T9g,  less  fractions  =  J. 


4fl  FOREIGN   MONEYS  OF  ACCOUNT* 

Three  Thalers  of  the  Convention  of  1838,  by  the  estimate  of  rela* 
tive  values  then  established  (gold  to  silver  as  14. 5G  to  1),  =  1  Ducat  j 
while  five  of  the  current  llixdollars,  established  in  1775,  by  the  then 
existing  estimate  of  relative  values  (gold  to  silver  as  14.483  to  1), 
=  1  Pistole,  or  10  guilder  piece,  above  referred  to. 

Most  of  the  moneys  of  account  throughout  Germany,  including 
Austria  and  Prussia,  are  based  upon  some  portion  of  the  fore- 
going. 

Foreign.  U.  Slates. 

Bremen :    5  schwaren  =*=  1  groot,  72  g.  or  2  florins  =■=  1 

Rixdollar  «  £  standard  Carl  d'or,  -  -  =  $0,781 

Frankfort :  4  pfennig  =  1  kreuzer,  60  k.  =»  1  Gulden  = 

tV  Convention  pistole,     *  -  -  -  »=    0.4022 

90  kreuzer  =  1  Rixdollar  of  account,  -        «=    0.0033 

1£  albus  =  1  kraiser-groschen,  30  k.  =  1  Rixdollar,   =     0.6033 
4  kreuzer  =  1  batzen,  22^  b.  mt  1  Rixdollar,  -  ==     0.6033 

120  kreuzer  or  2  gulden  or  1 J  Rixdollar  *a  1  Species 
Rixdollar;      -----«=    0.8044 
Hamburg,  Lubec :  12  pfennig  =»  1  schilling,  16  schillinge 

—  1  mark. 

1  mark    current  ■■  $0.28.     1   mark  banco,  -  =    0.35 
3  marks  banco  =  1  Specie  Rixdollar,    -            -        s=     1.05 

Baden.  —  Carlsruhe,  Heidelberg,  &c.  :  Same  as  Bavaria. 

Bavaria.  —  Augsburg,  Bamberg,  Bayreuth,  Munich, 
Nuremberg,  Ratisbon,  Wurtzburg ,  &e.  :  4  denari 
=  1  kreuzer,  60  k.  —  1  Florin,  -  -  -  =    0.4858 

2  kreuzer  =  1  albus,  2a.»l  batzen,  15  batzens 
=  1  Florin. 

Ik  florins  =  1  Rixdollar  current,  -  -        =*=    0.7287 

2  florins  =  1  Specie  Rixdollar. 

Hanover.  —  EmJcn,  Gottcngcn,  Hanover,  Osnaburg,  &c.  : 

12  pfennig  =  1  gute  groschen,  24  g.  s=a  1  Rixdollar,    =     0.7287 

lj  Rixdollar  ==  1  Specie  Rixdollar. 

60  kreuzer  =  1  gulden,  1|  g.  «■  1  Thaler,        -       =    0.094 

—  12  hellers  or  9   pfennig  =  1  albus,  32  albus 

=  1  Rixdollar,     -  -  -  -  -—     0.7281 

24  mari  a  =  1  reiuhsflorin,  l£  r.  =  1  Rix- 

dollar. 
60  kreuzer  =  1  golden,  L|  £  =  1  Thaler,        -         m     0.094 
iv,   Alton*,    Kul,  &c.  :    12  pfennig  =  1  schilling 
lubs,  lf>  ■ohillinge  [of  I^ilm-)  =  1  mark,     -        -  =s    0.35 

3  marks  =  1  Specieedaler  of  Denmark. 

Mi:<  KUNBDBO. —  Rostock,  Wismar,  &C.  :  Same  U  Han«>- 

Oldenhurg. — Same  as  Bremen;  also,  same  as  Ham- 
burg. 


FOREIGN   MONEYS  OF   ACCOUNT.  a  5 

Foreign.  U.  States. 

Saxony.  —  Dresden,  Leipsic,  &c.  :  12  pfennig  =  1  gute   • 

groschen,  24  g.  =  1  Species  Thaler,    -  =  $0.7287 

16  gute  groschen  =*=  1  reiehilorin,  2  r.  =*  1  Specie 
Kixdollar. 
Saxe.  —  Gotha,  Weimar :  Same  as  Hanover. 
Saxe  generally  and  Nassau  :  4  hellers  *■  1  kreuzer,  60 

kreuzers  =  1  Gulden,       -  -  -  -  am     0.4022 

l£  gulden  =  1  Rixdollar  current. 
Wurtemburg. — Halle,  Stuttgard,    Vim,  &c.  :    Same  as 

Saxe  and  Nassau. 
GREAT    BRITAIN.  —  Sterling  money:    Standard   for 
silver  coins  =  £  J  line  ;    for  gold  coins  ■»  \%  fine. 
Relative  values,  gold  to  silver  as  14.288  to  1. 
4  farthings  =  1  penny,  12  p.  =  1  shilling,  20  s.  = 

1  Pound  =  (4.866.*  U.  S.  Customs  value,       -     =    4.84 
GREECE.  — 12  denari  =  1   soldo,  20  s.  =  *  Lira   or 

Drachma,  -  -  -  -  -=     0.163 

HOLLAND. —  Standard  for  silver  coins  =  tnVu-  fine. 
Standard  for  gold  coins — Gouden  Willem  (10 
florins)  fractions  and  multiples  =  i9o0dp<j  fine  — 
Ducat  and  multiples  =  tVtfff  nn0  5  weight  of 
Ducat  =  3.494  grammes  am  $2.2827.  Relative 
values  —  gold  to  silver  as  15.604  to  1. 
100  centimes  =  1  Florin  or  Gulden  =  9.45  grammes 

of  fine  silver  =  145.843  troy  grains,         -  -=    0.3928 

2£  florins  =  1  Ryksdaalder,  =     0.982 

10  florins  in  gold  =  6.056   grammes  fine  gold  = 
$4.0251.   Custom-House  value  of  Florin  =  $0.40. 

India  and  Malaysia  or  East  Indies. 

British  Possessions.  —  Standard  of  purity  for  gold 
and  silver  coins  =  T£  fine,  each. 
Hindostan.  — Bombay,  Sural,  Tatta  :  100  reas  =  1  quar- 
ter, 4  q.  =  1  Rupee  =  165  troy  grains  of  fine 
silver,      -  -  -  -  -  -  =    0.4444 

*  For  all  time  since  1816,  the  government  of  Great  Britain  has  estimated  gold  and  silver 
as  14.2879  to  1.  The  pound  starling  of  mint  silver  weighs  1745.454  grains,  and  contains 
1014.545  grains  of  fine  silver.  The  value  of  the  pound  sterling  of  silver,  therefore,  rated 
by  the  United  States  standard  of  371}  grains  of  fine  silver  to  the  dollar.  Is  $4<849.  The 
value  of  the  silver  shilling,  of  full  weight,  is  $0.2174  'Hn;  pound  sterling  of  mint  gold 
weighs  123.274  grains,  and  contains  113.001  grains  of  fine  gold.  The  value  of  the  pound 
sterling  of  gold,  rated  by  t'ie  United  States  standard  of  23.22  grains  of  fine  gold  to  the 
dollar,  is  $4,866.  At  the  old  intrinsic  par  between  the  two  currencies,  viz.,  4  shillings 
and  6 pence  sterling  to  the  dollar,  or  $4,444  to  the  £,  the  par  of  exchange  is  9^  per 
cent.  Gold  is  the  money  standard  in  Great  Britain,  and  the  silver  coins  of  that  country 
are  not  legal  tender  at  home  in  sums  exceeding  £2. 
A* 


6*  FOREIGN   MONEYS  OP  ACCOUNT. 

Foreign.  U.  States. 

Calcutta :  12  pice  =  1  anna,  16  annas  =  1  Rupee,       «s  $0.4444 
1  Arcot  rupee  =  1  Sicca  rupee  *=  $0.44^f  f . 
1  Mohur  or  Gold  Rupee  »  165  troy  grains  fine  gold 

=  $7.1033. 
A  Lac  of  rupees  =  100.000  rupees. 
A  Crore  of  rupees  =  100  lacs  of  rupees. 
Madras.  — 12  pice  =  1  anna,  16  annas  =  1  Rupee,        =    0.4444 
80  cash  =  1  fanam,  42  fanams  =  1  Pagoda,     -        =    1.851 
3£  old  Sicca  rupees  =  1  Pagoda. 
1  Sicca  rupee  less  16  per  cent.  =  1  current  rupee. 
Cochin :  12  pice  =  1  anna,  16  annas  =  1  Rupee,  =     0.4444 

4  fanams  =  1  schilling,  5  schillings  =  1  Rupee. 
Goa  :  18  budgerooks  =  1  vintim,  4  v.  =  1  tanga,  5  t. 

=  1  Pardo,  -  -  -  -  m    0.96 

Pondicherry  :  60  cash  =  1  fanam,  24  f .  =  1  Pagoda,  =     1.84 
Banca  I.  —  Same  as  Batavia  {Java  I.) 
Borneo  I.  —  Borneo,  Pentaniah,  Sambos,  &c. :  Same  as 

Batavia  (Java  I.) 
Celebes  I.  —  Macassar:  Same  as  Batavia  (Java  I.) 
Ceylon  I.  —  Colombo :  4  pice  =  1  fanam,  12  fanams  = 

1  Ryksdaalder, =    0.3928 

Java  I.  —  Batavia,  Samarang,  Sarakarta :  5  duyt  =  1 
stuiver,  2  stuivers  =  1  dubbel,  3  dubbele  =  1 
schilling,  4  schillinge  =  1  Florin,  -  -  =    0.3928 

U.  S.  Customs  value  of  the  Florin  =  $0.40. 
Malacca.  —  Malacca :  4  duyt  =  1  stuiver,  6  stuivers  = 

1  schilling,  8  schillinge  =  1  Ryksdaalder,     -        =    0.795 
Molucca  Islands  or  Spice  I.  —  Same  as  Batavia,  Java  I. 
Philippine  I.  —  Luzon   I. — Manilla,  &c. :  34  mara- 

vedi  =  1  real,  8  reals  =  1  Peso,  -  -  =     1.00 

Siam.  —  Bangkok,  &c. :  4  tical  or  bats  =  1  tael,  100  t. 

=  lPecul, =    0.617 

Singapore  I.  — Same  as  Malacca. 
Sooloo  Islands.  —  Same  as  China. 
Sumatra  I.  —  Achcen:  4  copangs  =  1  mace,  4  m.  =1 

pardo,  4  p.  =  1  Tael,      -  -  -  -  =    4.1G 

Bencoolen :  8  satellers  =  1  soocoo,  4  s.  =  1  Real,        m     1.10 
IONIAN     ISLANDS.  —  Cephalonia,     Corfu,     Ithaca, 
Paxos,  Zante,  &c. :    12  denari  =  1  soldo,  20   s.  w 
1  Lira,  -  -  -  -  =** 

ITALY.  — 100  centesimi  =  1  Lira  Italiani,       -  -  =    0.187 

Ecclesiastical  States.  —  Standard  for  silver  coins  =- 
ffjjfinc;  for  goldoomm  ,'oVo  fiue\  Relative 
values,  gold  to  silver  as  15.526  to  1* 


FOREIGN    MONEYS  OP   ACCOUNT.  a  7 

Foreign.  U.  States. 

Ancona:  12  denari  =  1  soldo,  20  s.  =»  1  Scudo,   -        =  $1,007 

100  bajochi  or  80  bologni  or  10  paoli  =  1  Scudo. 
Bologna,  Ferrara :  12  denari  =  1  soldo,  20  s.  =  1  Lira,  =     0.201 
5  lire  or  10  paoli  or  100  baiochi  or  500  quattrini  = 

1  Scudo, =     1.007 

Rome :  5  quattrini  =  1  baiocho,  10  b.  =  1  paolo,  10  p. 
=  1  Scudo  =  TV  libbra  (373.78  troy  grains)  of 
fine  silver,      -  =     1.007 

Sequin  =  Tff  fine  =  T^  libbra  of  fine  gold. 
Lucca.  —  12  denari  =  1  soldo,  20  s.  =  1  Lira,        -        *m    0.243 
12  denari  =  1  soldo,  20  s.  =  1  Scudo  correnta,        =    1.007 
1  scudo  d'oro  of  Lucca  =  $1,127. 
12  denari  di  cambia  =  1  soldo  di  cambia,  20  s.  =  1 
Scudo  di  cambia,  -  -  -  -  =     1.068 

Lombardy  and  Venice.  —  Standard  for  silver  coins  =  ^ 
fine  ;  for  gold  coins  =  f  $  fine.     Relative  values, 
gold  to  silver  as  16  to  1. 
Bergamo,  Mantua,  Milan,  Padua,  Verona,  Venice,  &c.  : 

100  centesimi  =  1  Lira  Italiani,  -  =    0.187 

100  centesimi  =  1  Lira  Austriache,  -  -  =    0.162 

12  denari  =  1  soldo,  20  s.  =  1  Lira  correnta  di  Mi- 
lan =  -^f  jj  marcs  of  fine  silver  =  52.531  troy 
grains,  -  -  -  -  =    0.1415 

1  Lira  picola  di  Bergamo,  di  Verona  or  di  Venice,    =    0.098 
1  Lira  di  Mantua,   -  -  -  -  -  =    0.058 

Modena,  Parma.  — 100  centesimi  =  1  Lira,  -        =    0.187 

Naples.  —  Naples,    Salerno,   &c. :    7200    accini  =  360 
trapesi  =  12  oncie=  1  marco  or  libbra  =  4950.53 
troy  grains. 
Standard  for  gold  and  silver  coins  =  -^uutj  fine,  each. 
Relative  values,  gold  to  silver  as  15.373  —  to  1. 
10  grani  =  1  carlino,  10  c.  =  1  Ducato  =  ^  Nea- 
politan   libbra  of   fine    silver  =  295.55-f-  troy 
grains,  ----.=    0.796 

1|  ducati  =  1  scudo  or  crown. 
Sardinia  :  4608  grani  of  24  granottini,  each  =  1152 
carati  =  192  denari  =  8  oncie  =■  1  marco  =  3795 
troy  grains. 
Standard  for  gold  and  silver  coins  =  f  £  fine,  each. 
Relative  values,  gold  to  silver  as  15.545-[-  to  1. 
100  centesimi  =  1  Lira  Italiani,  -  =     0.187 

12  denari  =  1  soldo,  20  s.  =  1  Lira  di  Sardinia  =* 
*    ^V  marco  of  fine  silver  =  130.86  troy  grains,      -=    0.3525 
Genoa ;  5|  lire  fuori  banco  =  1  Pezza,    -  =    0.8854 


8tf  FOREIGN   MONEYS   OF   ACCOUNT. 

Foreign.  U.  States. 

4f  lire  di  cambio  =  1  scudo  of  Exchange,  -  -  —  $0.7187 

lOjVtf  lire  moneta  buona  =  1  Scudo  d'oro,     -  =     1.6663 

Tuscany. — Florence  —  moneta  buona:    12  denari  =  1 

soldo,  20  soldi  =  1  Lira,  =    0.1566 

12  denari  di  ducato  =  l  soldo  di  ducato,  20  s.  = 

1  Ducato  or  scudo  correnta,         -  -  -  =     1.0962 

7  lire  =  1  ducato.     7£  lire  =  1  scudo  d'oro. 
Leghorn.  —  moneta  lunga  :  12  denari  di  lira=  1  soldo 

di  lira,  20  soldi  di  lira  =  1  Lira,       -  =     0.15 

12  denari  di  pezza  =  1  soldo  di  pezza,  20  soldi  di 

pezza  =  1  Pezza,-  -  -  -  -=    0.9004 

6  lira  =  1  pezza. 

7  lire  moneta  buona  =  1  Scudo  corrente,  -        =     1.0962 
7£  lire  moneta  buona  =  1  Scudo  d'oro,        -  -  =     1.1745 

JAPAN. — Mats/nay,  Miaco,  Nangasaki,  Osaca,  Yedo,  &c. : 
10  cash  =  1  candarine,  10  c.  =  1  mace,  10  m.  =  1 
Tael,  _--_-=     1.4074 

MADEIRA  ISLANDS.— 1000  reas  =  1  Milrea,  -=    1.00 

MALTA  I.  —  20  grani  =  1  taro,  12  t.  =  Scudo,    -        =    0.406 
6  piccioli  =  1  carlino,  2  c.  =  1  taro,  12  t.  =  1 

Scudo, =    0.406 

2£  scudi  =  1  Pezza,  =     1.015 

MAURITIUS  I.  —  Port  Louis,  &c.  :  100  cents  =  1  Dol- 
lar, -  -  -  -  -  -=    0.9353 

20  sols  =  1  Livre  colonial  =  £  Franc  of  France,       =  0.09353 
10  livres  =  1  Dollar. 
MEXICO.  —  Acapulco,  Tampico,  Vera  Cruz,  &c. : 

6  grani  =  leuarto,  2  c.  =  1  medio,  2  m.  =  1  Real,    =     0.125 

8  reals  =  1  Peso,  -  -  -  -        =     1.00 
MOROCCO.  —  Fez,  Mogadore,  &c.  :  24  fluce  =  1  blan- 
ked, 4  b.  =  1  once,  10  onces  =  1  Mitkul,  =    0.7407 

NORWAY.  —  Bergen :  10  skillinge  =  1  mark,  6  m.  = 

1  Ryksdaler, =     0.5252 

2  Ryksdalers  =  1  Speciesdaler. 
Christiania,  &c.  :   12   pfennige=l  skilling,  6  s.  =  1 
ort,  4  orte  =  1  Ryksdaler  =  1  mark  banco  of 

Hamburg, =     0.3501 

.*>  Ryksdaler  =  1  Sj.r.irsdaler. 
PERSIA.  —  Bushirc,  &o.  :  5  denari  =  1  kasbeque,  10  k. 
=  1  Bhafree,  lis.  =  l  mainoode,  2  m.  =  1  abasse, 
50  a.  =  1  Toman,  =    2.233 

2  kasbequi  =  1  denaro-biste. 
PORTUGAL:  Standard  for  silver  coins  =  §£|  fine — for 
gold   ooinao*  \\   line.     Relative  values,  gold  to 
silver  as  15.3504-  to  1. 


FOREIGN   MONEYS   OF  ACCOUNT.  a  9 

Foreign.  U.  States 

Method  of    writing    and  reading  quantities:    Ex. — 
rs.  5  :  COO  0  750  =  5,000  inilreas  and  750  reas. 
l(i(IO   reas  a*  1    Milrea  =  the  silver  coroa  =  $%% 

mareo  of  lino  silver  =  415.435  troy  grains,      -      *=  $1,119 
1000  reas  =  1  Milrea  current — fluctuating,  about 

=  ,s().%. 
1  Milrea,  paper  —  fluctuating,  about  =  $0.81. 
480  reus  =  1  Crusado. 
PRUSSIA. — Standards  and  relative  values,   same  as 
given  under  Germany. 
Berlin,   Brandenburg,  Dantzic,  Potsdam,  Magdeburg, 
Stetin,  &c.  : 
12  pfennig  =  1  gute  groschen,  24  g.  =  1  Rixdollar 

or  Thaler  current  =  lgV  specie  thaler,  -        =    0.7287 

H  Rixdollar  =  1  Thaler  banco  =  $0.91HV 
1$  florins  —  1  Thaler  specie,  -  -  -  =     0.694 

Cologne:  12  hellers  =  1  albus,  80  a.  =  1  Rixdollar 

=  23(j  Convention  pistole  =  1£  florins  d'or,   -        =    0.6033 
78  albus  =  1  Rixdollar  current. 
120  fettmangen  =  90  kreuzer  =  30  groschen  =  20 
blafferts  =  3£  Cologne  florins  =  2  heron  florins 
=  l£  rader  florins. 
Aix  la  Chapelle,  Crevelt,  Elberfeldt,  &c.  : 

4  pfennig  =  1  kreuzer,  60  k.  =  1  florin,  1|  f.  =  1 

Thaler,  ...-.=    0.694 

Brunswick:    8   pfennig  =  1  marien-groschen,  36  m. 

=  1  Thaler, =     0.694 

Konigsberg :   6  pfennig  =  1  schilling,  3  s.  =  1  gros- 
chen, 30  groschen  =  1  florin,  3  f .  =  1  Thaler,       =    0.694 
8  specie  gute  groschen  of  Berlin  =  30  groschen  of 
Konigsberg. 
PRINCE  OF  WALES  I.  —  10  pice  =  1  copang,  10  c. 

=  1  Dollar, =     1.00 

PROVINCES  of  New  Brunswick,  Nova  Scotia,  New- 
foundland, AND  THE  CANADAS : 
4  farthings  =  1  penny,  12  p.  =  1  shilling,  20  s.  = 

1  Pound, =    4.00. 

RUSSIA.  —  Standard  for  silver  coins  =  £  fine,  for  gold, 
coins  =  -fj  fine.     Relative  values,  gold  to  silver 
as  15^$-  to  1. 
Archangel,  Cronstadt,  Helsingfors,  Odessa,  Revel,  Se- 
vastopol, St.  Petersburg,  &c.  : 
10  kopecs  =  1  grieven,   10  g.  =  1  Rublyu   (ruble) 

=  B3ff  funt  of  fine  silver  =  278.47  troy  grains,      =    0.75 


10  a  FOREIGN   MONEYS  OF  ACCOUNT. 

Foreign.  U.  States. 

2  denushkas  or  4  polushkas  =*  1  kopec.     33  J  altins 
=  1  ruble. 
Riga.  —  Same  as  St.  Petersburg,  also  — 

30  groschen  =  1  florin,  3  f.  =  1  Kixdollar.    1  Alber- 

tus  dollar,  -  -  -  -  -  =  $1.00 

SARDINIA  I.  —  100  centesimi  —  1  Lira  Italiani,  =    0.187 

1  Lira  di  Sardinia,         -  =    0.354 

SICILY    I.  —  Standards  of  purity  and  relative  values, 
same  as  Naples. 
G  picioli  =  1  grano,  20  g.  =  1  taro,  30  t.  =  1  Oncia 

=  £f  Neapolitan  libbra  of  fine  silver,     -  -  =    2.388 

8  picioli  =  1  ponti,  15  p.  =  1  taro,  10  t.  =  1  ducato. 
6  tari  =  1  fiorino  or  florin,  2  f .  =  1  scudo,-  2£  s.  = 

1  oncia. 

SPAIN.  —  Standard  for  silver  coins  since  1786,  peso  and 
£  peso  =  y§  fine  ;  peseta,  real  and  £  real  =  f  | 
fine ;  for  gold  coins  =  %  fine,  except  the  coronilla 
(gold  dollar)  =  Tf  $  fine.  Relative  values,  since 
1786,  gold  to  silver  as  16.39  to  1. 
Real  velldn  =  ^V  peso  duro,  -  -  -  =    0.05 

Real  de  plata  nuevo  =  y1^  peso  duro,    -  =    0.10 

Real  de  plata  Mexicana  =  $  peso  duro,       -  -  =    0.125 

Real  de  plata  antiquas  =  j^5  peso  duro,  -        =    0.0943 

Real  d'Alicant  =  j3?2F  peso  duro,    -  -  -  =    0.0754 

Re£l  de  Valencia  =  ■££■$  peso  duro,      -  =    0.0566 

Real  currante  de  Gibraltar  =  T^-  peso  duro,  -  =     0.0835 

Real  de  Catalonia  =  5V5  Peso  duro,     -  -        =     0.0808 

Real  ardita  de  Catalonia  =  -5^  peso  duro,  -  =    0.0539 

8  reale  de  plata  antiquas  =  1  Piastre  or  peso  of  ex- 
change =  g£  peso  duro,         -  -  =    0.7543 
Peso  duro  =  y1^  marco  of  fine  silver  =  371.9  troy 

grain*,     -  -  -  -  -  =    1.0018 

40  dineri  =  16  comadi  =  8  bland  =  4  maravedi  = 

2  ochavi  =  1  quarto,  4$  quarti  =  1  sualdo,  2  s. 
=  1  Real. 

Alkaut :  34  maravedi  =  1  real,  10  r.  =  1  Piastre,       =    0.7543 
Barcelona,    Tortosa^2  malli »  idinero,    1 12  <1.  =  1 
BOaldo,   20  s.  =  1   Libra  =  10   reule   ardita  de 
Catalan,  -  -  -  -  -  =     0.5388 

liilboa,     Carlhagcna,    Madrid,    Malaga,     Santander, 
Toledo: 
34  maravedi  =  1  real,  15  r.  =  1  Peso  sencillo. 


FOKEION   MONEYS  OS  ACCOUNT.  all 

Foreign.  U.  States. 

15jV  reiile  =  1  peso  de  plata  or  Piastre,  -        =  $0.7543 

4  piastres  =  1  doubloon  de  plata  or  pistole  of  ex- 
change. 
Cadiz,  Sevilla :  34  maravedi  or  16  quarti  =  1  Real,      =     0.0943 

8  mile  =  1  Piastre,  10|  reale  =  1  Peso  duro. 
Gibraltar :  34  maravedi  =  1  real,  9  r.  =  1  Piastre,      =    0.7543 

12  mile  =  1  Peso  duro. 
Valencia :  12  dineri  =  1  sualdo,  2  s.  =  1  real,  10  r. 

«  1  Libra, =    0.5657 

lj-  libra  =  1  Piastre  or  peso  of  account,     -  -  =    0.7543 

24  diueri  =  1  real,  10  r.  =  1  Peso  duro,  -        =     1.0018 

SW  EDEN.  —  Standard  for  gold  coins  (ducats,  multiples 
and  fractions)  =  f  fs  nne  i  f°r  silver  coins  =  %fy 
fine.    Kelative  values,  gold  to  silver  as  14.692  to  1. 
Carlscrona,  Gefle,  Gottenburg,  Stockholm,  &c. : 

12  rundstycken  or  ore  =  1  skilling,  48  s.  =  1  Riks- 
daler  =  ifg  mark  of  fine  silver  =  393.68  troy 
grains,      -  -  -  -  -  -=     1.0604 

100  centimes  or  skillings  =  1  Riksdaler,  -        =     1.0604 

SWITZERLAND.  —  1  Livre  de  Suisse,  of  the  convention 
of  1814,  =  l£   litres  tournois  of  France  =  f^ 
francs  of  France,  -  -  -  -  =    0.2771 

Berne,  Basle,  Lausanne,  Lucerne,  Pay  de  Vaud : 
10  rappen  =  1  batz,  10  batzen  =  1  Livre  de  Suisse. 
12  deniers  =  1  sou,  20  sols  de  Suisse  =  1  Livre. 
10  rappen  =  1  batz,  15  b.  =  1  Florin  or  Guilder,     m$    0.4156 
8  hellers  =  1  kreuzer,  60  k.  =  1  Florin. 
Geneva  :    12  deniers  =  1  sou,  20  s.  =  1  Livre  =  3£ 

florins  petite  monnie  =  1%  francs  of  France,  =    0.3117 

3  livres==  1  Ecu  or  Patagon,  -  *m    0.8313 

ISeufchatfrr  100  rappen  =  1  Franc  or  Livre  de  Suisse. 
12  deniers  =1   sol,   20   s.  =  1    Livre   tournois   de 

Neufchatel  =  2£  livers  foible,      -  -  -  =    0.2628 

St.'  Gaul:  480  heller  =  240  pfennig  =  60  kreuzer  = 
15  batzen  =  10  skilling  =  1  Florin  or  Guilder. 
1  florin  current  =  2£  francs  of  France,  -        =    0.4365 

1  florin  specie,         r  -  -  -  -  =    0.5187 

Zurich:  60  kreuzer  of  8  hellers  each,  or  16  batzen  of 
10  augsters  each,  or  40  skillings  of  12  hellers  each 
=  1  Guilder  or  Florin,  =    0.4365 

TRIPOLI.  — 100  paras  =  1  Piastre  or  Ghersch,  ghersch 

of  1832, =    0.10 

TUNIS.  —  Tunis,  Biserta,  Susa,  &c. :   2  burbine  =  1 


12  a  TORSION   MONEYS  Off  ACCOUNT, 

Foreign.  U.  States. 

asper,  52  a.  or  16  carobas  =»  1  Piastre,*  piastre 

of  1838, =$0,128 

TURKEY.  —  Constantinople :  3  aspers  =  1  para,  40  p. 
=*  1  Piastre  or  Ghersch.* 
With  the  Dutch,  French  and  Venetians,  100  aspers  =  1 

Piastre. 
With  the  English  and  Swedes,  80  aspers  =  1  Piastre. 
500  piastres  =  1  chise  ;    30,000  piastres  =  1  kitz  ; 
100,000  piastres  =  1  juck. 
Smyrna :  40  paras  or  medini  =  1  Piastre  or  Gooroosh. 
12  tomans  =  1  Piastre  or  Gooroosh. 

West  Indies. 

Cuba  I.  —  Cardenas,    Cienfuegos,    Havana,    Matanzas, 
Mariel,  Nuevitas,  Porto  Principe,  Sagua  laGrande, 
St.  Jago,  &c.  : 
34  maravedi  =  1  real,  8  r.  =  1  Peso,  -  -  =     1.00 

Hayti  I.  —  Aux  Cages,  Cape  Haytien,  Port  au  Prince, 
San  Domingo,  &c.  : 
100  centesimi  =  1  Dollar  or  Peso  duro  =  11  eseu- 

lini,  .-_--=     1.00 

1  dollar  Haytien  currency  =  7J  esculini,    -  -  =    0.66 

Porto    Rico   I.  —  Guayama,     Mayaguez,    St.    Johns, 
Ponce,  &c.  : 
34  maravedi  =  1  real,  8  r.  =  1  Peso,  -  -        =1.00 

British  Islands. — Anguilla,  Antigua,  Barbuda,  Do- 
minica, Grenada,  Montscrrat,  Nevis,  St.  Kitts,  St. 
Lucia,  St.  Vincent,    Tobago,    Tortola,    Trinidad, 
Virgin  Gorda: 
4  farthings  =  1  penny,  12  p.  =  1  shilling,  20  s.  = 

1  pound,  -  -  -  -  -  =    2.222 

Nassau  and  the  Bahamas  generally : 
4  farthings  =  1  penny,  12  p.  =  1  shilling,  20  s.  = 

1  Pound, =    2.485 

Pound  of  Turks  I.    -  -  -  -  -=     3.00 

Barbadoks  I.  —  Bridgetown,  &u.  :  1  Pound,  -         =     3.20 

Jamaica  I.  —  Falmouth,  Kingston,  Morant  Bay,  Savan- 
nah la  Mar,  &c.  : 
4  farthings  =  1  penny,  12  p.  =  1  shilling,  20  s.  = 

1  Pound, =    3.00 

*  The  coins  of  tin-  Turkish  government,  owing  to  frequent  and  oft-repeated  deterioration 
by  enactments,  have  no  definable  standard  value  whatever.  Bills  of  exchange  on  Turkey 
arc  usually  drawn  in  Spanish  dollars.  The  rains,  of  the  lOrer  piastre  "t  Turkey,  of  full 
w.-ixht,  of  1775,  is  $0,446  ;  of  that  minted  it.  Tunis  in  1787,  $0,259  ;  of  that  of  Turkey  of 
1818.  $0,182,  and  of  that  of  1836,  $0,128,  while  that  issued  only  a  few  years  since,  id 
worth,  intrinsically,  but  about  4  cents. 


FOREIGN    MONEYS   OF   ACCOUNT.  ft  13 

Foreign.  U.  States. 

Danish  Islands.  —  Santa  Cruz,  St.  John,  St.  Thomas, 
St.  Bartholomew  : 
12  skillings  =  1  bit,  8  b.  =  1  Ryksdaler,        -        =»  $0.64 
100  cents  =  1  Ryksdalor. 
12£  bits  =  1  Spanish  dollar. 
Dutch  Islands.  —  Saba,  St.  Eustatius,  St.  Martin: 

6  stuivers  =  1  redl,  8  r.  =  1  Piastre,  -  -  =    0.73 

11  reiils  or  Esculins  =  1  Spanish  dollar. 
French  Islands.  —  Deseada,    Guadeloupe,   Mariega- 

lante,  Martinique: 

12  deniers  =  1  sol,  20  s.  =  1  Livre  =  §  livre  tour- 

nois,  ------        a    0.1232 

4  farthings  =  1  penny,  12  p.  =  1  shilling,  20  s.  = 

1  Pound,  -  -  -  -  -  =    2.222 

Little  Antilles,  generally,  Same  as  Mexico. 
B 


14 


FOREIGN  LINEAR  AND   SURFACE  MEASURES 
REDUCED   TO   UNITED   STATES. 

Foreign. 
ABYSSINIA.  —  Massuah  :  8  robl=  I  derah  or  pic, 
ALGIERS.  — 10  decimetres  =  1  metre,      - 

8  robi  =  1  pic.     Pic,  Moorish,  for  linens,   - 

Pic,  Turkish,  for  silks,  &c,  - 
ARABIA .  —  1  kassaba  =  12.31  ft.     Mile, 
Aden  :  8  robi=  1  yard  or  pic,      - 
Jidda  :  8  robi  =  1  pic,  • 

Mocha :  8  gheria  =  1  covid.     Covid  (land), 
Covid  (for  iron,  <J*c-)> 
8  robi  =  1  gez,  - 

AUSTRIA.  —  (Imperial,  or  legal  and  general)  : 
Vienna,  Trieste,  Prague,  Lmtz,  $c.  : 

12  7011=1^        -  - 

29£zoll  =  lelle,         - 

6  fus  =  1  klafter,  4000  k.  =  1  meile, 

10fus=lruth  (builders,)1    - 

3  metzen  =  l  joeh,  - 
(Special  and  local)  — 
Upper  Austria.  — Lintz,  dfc. :  1  elle,    - 
Bohemia.  —  Prague,  4fc. :  2  fus —  1  elle, 

4  elle  =  1  dumplachter,  ... 
Hungary.  —  1  fus  =  1.037  feet.     1  elle, 
Moravia. — 2§  fus  =  1  elle,        ... 

AZORE  ISLANDS.  —  Same  as  Lisbon  (Portugal). 
BALEARIC  ISLANDS.  — 3  pie  or  4  palma  =  l 
vara,  2  vara  =  l  cana. 
Majorca.  —  1  cana, 
Minorca.  —  1  cana,         - 
BELGIUM.  — 10  streep  =  1  duhn,  10  d.  =  1  palm, 
10  p.  =  1  el.  =  1  metre  of  France, 
10  el  =  1  roed,  100  r.  =  1  mijl, 
2£  fus  =  1  aune. 
Antwerp  :  Aune  for  cloths,     - 

.    Aune  for  silks,  - 

Brussels:  1  aune =0.761  yards.    Vaem,     -       =    2.- 


U.  i 

States. 

0.682 

yard . 

1.094 

i< 

0.519 

M 

0.092 

(( 

1.22 

miles* 

0.95 

yard. 

0.743 

a 

1.58 

feet. 

2.25 

u 

0.694 

yard. 

1.037 

feet. 

0.852 

yard- 

4.712  miles- 

L0.37 

feet. 

1.422 

acres* 

0.874 

yard. 

0.65 

i« 

2.598 

it 

0.874 

it 

0.865 

u 

1.711 

yards. 

1.754 

M 

1.003 

yards. 

0.621  "mile. 

0.749 

yard. 

0.761 

M 

FOREIGN   LINEAR   AND  SURFACE  MEASURES.  a  15 

Foreign.  U.  States. 

Mechlin:  1  aun<\  -  -  -  -=    0.753  yard. 

BERMUDAS  I.  —  Same  M  Gkeat  Britain. 
BOURBON  I.—  3  pied  =  1  aune,        -  -       —     1.298     " 

BRAZIL.  —  12  pollegada  =  l  pe,  5  pes  =  1  passo, 
52  passi  =  1  cstadio,  24  estadi  =  1  milha,  3 
milhe  =  1  legoa,       -  -  •  -«=    3.836  miles. 

8  pollegada  =  1  palmo,  5  palmi  =  1  vara,  2 

vare,  or  3^  covadi,  or  1£  passi  =  1  braca,    =    7.214    feet. 

1  geira,      .----=     1.428  acres. 
Bahia,  Rio  Janeiro;  3  palmi  =  1  covado,  -=    0.713  yard. 

Central  and  South  America. 

Balize,  Bolivia,  Buenos  Ayres,  Chili,  Equador, 
Guatimala,  New  Granada,  Peru,  Uruguay, 
Venezuela,  Yucatan  : 
Nomenclatures  and  legal  values,  same  as  Castile 

( Spain). 
Guiana.  —  Berbice,  Demerara,  Essequibo,  Surinam: 

Same  as  Holland. 
Cayenne.  —  Same  as  France. 
CANARY  I.  —  12  onza=  1  pie,  3  p.  =  1  vara,      =    0.920  yard. 

2  vara  =  1  braza,         -  -  -  -=     5.522    feet. 
52  braza  cuadrada=l   celemin,  12  c.=  l  fa- 

negada,  -  -  -  -  =    0.5       acre. 

CANDIA  I.  —  8  robi  =  l  pic,      -  -  -=    0.697  yard. 

CAPE  COLONY.  —  Same  as  Great  Britain. 
CAPE  VERDE  I.  — Same  as  Lisbon  (Portugal). 
CHINA. —  10  fan=l  tsun  or  punt,  10  tsun=l 
kong-pu  or  chik,  10  kong-pu  =  1  cheung, 
10  cheung  =  1  yan,  18  yan  =  1  li,   -  -  =    0.346    mile. 

Chik  (mathematical)  =  1.094  ft.      Chik    (en- 
gineers'), -  -  -  = 
Chik  (tradesmen' s)  =  1.218  ft.     Kong-pu,       -  = 
l£    chik    (engineers'1)  =  1    thuoc,   3£   thuoc 

=  1  po,  -  -  -  -  = 

10  punts  =1  covid  or  cobre,  If  c.  =  1  thuoc 

{mercers''),     -  -  -  -  = 

Pekin  :  10  chik  (math.)  =  1  cheung,  -  -        = 

CYPRUS  I.  —  8  robi=  1  pic,  -  -  -  = 

DENMARK.  —  24  tomme  or  2  fod  =  1  aln,       -       = 

3  aln  =  1  favn,  If  f.  =  1  rode,  2400  r.  =  1  miil,  = 

96  album  or  8  skiepper  =  1  toende,    -  -  = 

EGYPT.  —  2  derah  =  1  fedan,  3  f .  =  1  gasab,        = 

8  rob  =  1  pic,         -  -  -  =* 


1.058 
1.014 

feet. 

n 

5.025 

it 

0.711 

10.937 

0.696 

0.688 

yard. 

feet. 

yard. 

4.681  miles. 

5.45    acres. 

12.67      feet. 

0.74    yard. 

16  a  JOREIGN    LINEAR   AND   SURFACE   MEASURES. 

• 

Foreign.  U.  Slates. 

1  fedan  al  rieach,  -  -  -  -  =    4. acres 

Alexandria,  Rosctta:  1  pic  stambuli,              -        =  0.733  yard. 

Pic  for  muslins,  &c,           -  =  0.686      " 

Pic  for  cloths,        -        -      =  0.613      M 
FRANCE.  —  100  centimetres  or  10>  decimetres  =  1 

metre,      -            -            -            -                -  =  1.094     " 

100  metres  or  10  decametres  =  1  hectometre,  =  19.883  rods. 
100  hectometres  or  10  kilometres  =  1  myria- 

metre,      -            -            -            -                  =  6.214  miles. 

100  square  metres  =  1  are,  100  a.  =  1  hectare,  =  2.471  acres. 

3T6^  pied  metrique  =  1  aune  =  47 £  inches,      -  =  1 .312  yards. 
GERMANY.—  Baden  {legal)  :  20  zoll  or  2  fus  = 

lelle,                                                              =  0.656     " 

5  elle  =  1  ruthe =  3  metres  of  France,        -      =  3.281     " 

2  stunden  =  1  meile,  -  -  -  -  =  5.524  miles. 
1  jauchart  =  0.82  acre.     1  morgen,           -        =  0.889   acre. 

Manheim  :  1  fus  =  0.952  ft.     1  elle,       -            -  =  0.610  yard. 

Bavaria  (legal)  :  120  zoll  or  10  fus=l  ruthe,     =  9.575     feet. 

2400  ruthe  =  1  meile,                                          =  4.352  miles. 

34^  zoll  =  lelle,          -                        -            -=  0.911  yard. 

1  jauchart  or  morgen,         -            -                     =  0.841    acre. 

5  cubic  fus=  1  klafter  =  110.62  cubic  feet. 

Augsburg:  2  fus  =  lelle,            -            -            -=  0.648  yard. 

1  elle  (mercers'),  -  -  =  0.666  " 
Nuremberg :  2$  fus  =  1  elle,  -  -  -=  0.718  " 
Hanover  (legal)  :  12  zoll  =  l  fus,              -         =  0.943    foot. 

2  fus  =  1  elle  =  0.638  yard.  8  e.  =  1  ruthe,  =  15.328  feet. 
1462£  ruthe  =1  meile,  -  -  -  =  4.246  miles. 
2  vierling  =  1  morgen,       -            -                     =  0.647    acre. 

Bremen  :  24  zoll  or  2  fus  =  l  elle,           -            -  =  0.633  yard. 

6  fus=  1  klafter,  2f  k.  =  l  ruthe,  -  =  15.188  feet. 
20000  Rhineland  fus  =  1  meile,  -  -  =  3.896  miles. 
120  square  ruthe  =  1  morgen,  -  =  0.636  acre. 
1  reif  =  96.52  cub.  ft.    1  faden  =  61.6  cub.  ft. 

Emdcn,  Osnaburg  :  2{  fus  =  lelle,         -             -=  0.(V.»S  yard. 

Hesse  Cassel.  —  24  zoll  or  2  fus  =1  elle,      -       =  0.623      " 

14  fus  =  l  ruthe,        -  -  -  -=13.088     feet. 

lklafter  =  126.089  cubic  feet. 
Hesse  Darmstadt  (legal)'.  100  /.oil  or  10  fus  = 

1  klafter  =  2£  mUrcs  of  France,             -        =  8.202     feet. 

32  zoll  =  1  elle,          -           -           -           -=  0.875  jard. 

400  square  klafter  or  4  viertel  =  1  morgen,        =  0.618    acre. 

Frankfort:  12  zoll  =  1  fus  or  werksi-huh,      -         —  0.934     foot. 

2fus=l  elle,  2e.  =  l  stab,     -            -            -  =  1.245  jorae. 

10  feldfus=l  ruthe,        -  -  -         =11.672    feet. 


FOREIGN   LINEAR  AND  SURFACE  MEASURES. 


«17 


Foreign. 
Holstein.  —  Hamburg,  Altona  : 

24  zoll  or  6  palm  or  2  fus  a  1  ello,     - 
3  ello  =  1  klafter,  2\  k.  =  1  marschruthe, 
2§  klafter  (16  fus)  =  1  geestruthe, 
24000    Rhineiand    fus  (2000    R.   ruthe)  «=  1 
meile,  -  -  -  -  - 

1  Brabant  elle  for  woollens, 
COO  square  marschruthe  =  1  morgen, 
Lubec :    Denominations  and  relative  values,  same 
as  at  Hamburg.  —  1  elle,  - 

Mecklenburg.  —  Rostock,  dfc. —  Same  as  Ham- 
burg. 
Saxony.  —  Dresden,  Leipsic:  12  linie  =  1  zoll,  12 

zoll  =  1  fus,  2  fus  =  1  elle, 
•    2  elle  =  1  stab,  4  stab  =  1  ruthe,        -  -  i 

3  elle  =  1  klafter,  - 

1500  ruthe  =  1  meile,  - 

300  square  ruthe  =  1  acker, 
Freyburg  :  2  fus  =  1  elle,  5  e.  =  1  ruthe, 
Oldenburg.  —  24  zoll  or  2  fus  =  1  elle, 

9  elle  =  1  ruthe,  1850  r.  =  1  meile,     - 
GREAT  BRITAIN.  —  Same  as  United  States. 
GREECE.  —  Patras  :  8  robi  =  1  pic  for  silks, 

1  pic  for  woollens,  dj^c., 
HOLLAND  (legal)  :  10  streep  =  1  duim,  10  d.  = 
1  palm,  10  p.  =  1  el  =  1  metre  of  France, 

10  el  =  1  roed,  100  r.  =  1  mijl, 
Previous  to  1820  —  2&  fus=  1  el,    - 

El  of  Flanders,  - 

Hague  —  Brabant  el,  - 


U.  States. 

m  0.G266  yard. 
=  13.159  feet. 
=    5.013  yards. 

=  4.68  miles. 
=  0.761  .yard. 
—    2.385  acres. 

=    0.63    yard. 


0.618 
4.943 
5.561 
4.213 
1.515 
9.619 
0.648 
6.133 


yard. 

M 

feet, 
miles, 
acres. 

feet, 
yard, 
miles. 


0.694  yard. 
0.75       " 


1.093 
0.621 

0.747 
0.776 
0.761 


mile, 
yard. 


India  and  Malaysia  or  East  Indies. 

An-nam.  —  Same  as  China. 

Birmah.  —  4  taim  =  1  sadang,  7  s.  =  1  bambou,    =  4.208  yards. 

Ceylon  I.  —  Colombo  :  5  palmi=  1  covid,       -        =  0.516      " 
Hindostan.  —  Bombay  :  2  tussoo  =  1  gheria,  8  g., 

=  1  haut  or  covid,  -            -            -            -  =  0.503      " 

1  J- haut  =  1  guz,                                                  =  0.755      " 
Calcutta  :  3  jaob  =  1  angulla,  3  a.  =  1  gheria, 

8  g.  =  1  haut  or  covid,  2  h.  =  1  ghes  or  guz,  =     1. " 

3  palgat=  1  hand,  5  h.  =  l  cubit,      -            -  =  1.25       foot. 

3  cubits  =  1  corah,  1728  c.  =  1  coss,        -        =  1.227  miles. 

Goa:  12pollegada=lpe,         -           -            -  =  1.082    feet. 

B* 


18  a 


EOBEIGN   LINEAR   AND   SURFACE  MEASURES. 


Foreign. 

24f  pollegada  =  1  covado  avantejado,        -       = 

13^  pollegada  =  1  tcrca,  3  t.  =  1  vara,  -  = 

Madras :  8  gheria  em  1  covid,  -  = 

1  cassency  or  cawney,  -  -  -  = 

Massulipatam,  2  palm  =  1  span,  3  8.  =  1  cubit,  = 

Mysore,  Sringapatam :  8  gerah  =  1  haut,  2  h. 

=  1  gugah,  = 

Pondiclv  rry  :  8  gheria  =  1  haut  or  covid,  -  = 

Sural :  84  tussoo  or  20  wiswusa  =  1  wusa,  = 

18  tussoo  =  1  cubit  or  haut/or  matting,    -       = 
Tatta  :  10  garca  —  1  guz,  -  -  = 

Tranquebar,    Serampore    (legal)  :  same  as  Den- 
mark, 
Java  I.  — Batavia  :  8  gheria  =  1  covid,  cubit  or  el,  = 
1  fus  (Rhenish),     -  -  = 

Malacca.  —  Malacca  :  8  gheria  =  1  covid,  -  -  = 

8  covid  =  1  jumba,  -  -  = 

Philippine  I.  —  Luzon  J.  —  Manilla.  —  Same  as 

Cadiz,  Spain. 
Siam.  —  12  nion  =  1  keub,  2  k.  =  1  sok, 

2sok=l  ken,  2  k.  =  1  vouah,  20  v.  =  l  sen 
40  s.=  1  jod,  25  jod=  1  roeneng, 
Bangkok  :  -8  gheria  =  1  covid,     - 
Singapore  I. —  8  gheria  =  1  hasta, 
Sumatra  I.  —  4  tempoh  or  2  jankal  =  1  etto,  2  etto 

=  1  hailoh, 
IONIAN  ISLANDS.  —  Ccphalonia,  Corfu,  Illiaca, 
Paxos,  St.  Maura,  Zantc,  &c.  : 
12  onue  =  1  pie,  5  pes  =  1  passo, 
1  braccio  for  silks,        - 
1  braccio /or  woollens,        - 
1  moggio  (linear) , 
30  inches  or  3  feet  =  1  yard, 

ITALY. —  LOHBAKDY    AM>   VENICE  : 
Government  and  Customs  Measure  — 
10  atome=  1  dito,  10  d.  =  1  palmo,  10  p.  =  1 

metro  or  braccio  —  1  metft  of  FranoOj 
1000  metre  =  1  miglio,       - 
100  square  metre=l  tavnla,  100  t.  =1  torna- 
tura,  ----- 

Special  and  local  — 

Venice:  2  palmi=  1  braccio,  2£  b.  =  1 
1£  passi  =  1  pertica. 
44  pede  —  1  chebbo,  l£  c.  =  1  QMttBO 
1  passo,  geometrical, 
1  braccio  for  woollens, 


U.  States. 

0.744 

yard. 

1.203 

M 

0.515 

« 

1.32 

acres. 

1.594 

feet. 

1.072  yards. 

0.5 

ii 

2.712 

u 

0.581 

ii 

0.943 

M 

0.75 

II 

1.03 

feet. 

0.5 

yard, 

12.— 

feet. 

-=    0.525  yard. 

2.388  miles. 
1.5  feet. 

1.5 

-=     1.08     yard. 


5.455     feet. 
0.705  yard. 
0.7.M      » 
2.4     miles. 

1. yard. 


3.281     feet. 
0.621    mfle. 

12.  iT  1  acres. 

5.099    feet. 

5       » 
0.73'J       " 


FOREIGN    LINEAR   AND    SURFACE  MEASURES.  a   19 

Foreign.  U.  States. 

1  braccio,  for  silks,  -  -  =  0.693    feet. 

Naples.  —  5  minuto=  1  oncia,  12  o.  =  1  palmo,  8 

palrni=  1  carina,      -  -  -  -  =  6.92         " 

7£  palnii  =  l  passo  or  pertica,  8  pertica=l 

catena,  11 6|  catene=  1  miglio,  -        =  1.147  milea. 

900  square  passi  =  1  moggio,   -  -  -  =  0.87      acre. 

1  braccio  (2§  palmi  in  theory),       -  =  0.764  yard. 

Sardinia.  —  Genoa :  8  oncie  =  I  pie,  10  p.  or  12 

palmi  =  1  canna  (surveyors'),  -  -  =  9.715     feet. 

2 J  palmi  =  1  braccio,        -  -  -        =0.63     yard. 

9  palmi  =  1  canna  picolo,        -  -  -  =  2.429      " 

10  palmi  =  1  canna, for  linens. 
12  oncia  =  1  pie  liprando. 

Nice :  12  oncia  =  1  palmo.     25  oncia  =  1  raso,     =    0.600      " 
Turin;  8  oncia  =  1  pie  manual,        -  -        =1.19       feet. 

12  oncia  =  1  pie  liprando. 

14  oncia  =  1  raso,        -  -  -  -=    0.649  yard. 

5  pie  manual  =  1  tesa. 

6  pie  lip.  =  1  trabucco,  2  t.  =  1  pertica. 

States  of  the  Chruch. —  Ancona:  1  braccio,        =    0.704      " 
10  pie  =  1  pertica,  -  -  -        =13.438    feet. 

Rome:  10  decline  or  5  minuto=  1  oncia,  16  o. 

=  1  pie,  5  piede  =  1  passo,  -  -  =    4.884    feet. 

5  linea  =  1  parto,  24  p.  =  1  palmo,  8  palmi = 

1  canna,  -  -        =    2.176  yards. 

Tuscany.  — Leghorn,  Florence,  Pisa: 

12  denari  or  3  quattrini  =  1  soldo,  20  soldi  or  2 

palmi  =  1  braccio,  4  b.  =  1  canna,  -  -  =     2.552     '* 

8  braccia=l  passo,  2  p.  =  l  cavezzo,      -       =  11.484  feet. 

5  braccia=  1  pertica,  566J  p.  =  1  miglio,       =     1.027  miles. 

JAPAN. — 5  Kupera  sasi  =  1  ink,  =     2.072  yds. 

l£  sasi  =  lk.  sasi,  2\  sasi  =  1  ikje,  -      -       =      2.32  feet. 

MALTA  I.  —  12  oncie  =  1  palmo,  8  palmi  or  7 J 

piede  =  1  canna,      -  -  -  -  =    2.275     " 

MADEIRA  I.  —  Standard  same  as  Lisbon,  Porl- 


MAURITIUS  I.  —  Standard  same  as  Great  Brit- 
ain. 
MEXICO.  — Same  as  Cadiz,  Spain. 
MOROCCO.  —  Mogadore:  1  cadee,      -  -        =     1.695     feet. 

1  covado  =  1.654  feet.    1  pic,  -  -=    0.723  yard. 

NORWAY.  —  Same  as  Denmark. 
PERSIA.  — 1  archiii- ariscli,     -  -  -        =1.063      " 

1  arc-bin  schah,  -  -  -  -  =     0.874      " 

1  gueza  (royal)  =  3T\y  feet.      1  gueza  (com- 
mon),     =    0.692      " 


20a 


FOREIGN   LINEAR   AND  SURFACE  MEASURES. 


Foreign. 
1  monkelzer,    -  -  -  -  -  = 

Bushire :  1  guz,  = 

PORTUGAL.  —  Lisbon,  St.  Ubes,  fc. : 

80  pollegada,  or  10  palmi  do  craveira,  or  6§ 
pes,  or  3  J  covado,  or  2  vara,  or  1 J  passo  =  1 
braca,  -  -  -  -  -  = 

780  pes  =  l  estadio,  8  estadi  =  l  milha,    -        = 
3  milha  =  1  legua. 

8  pollegada  =  1  palma,  3  p.  =  1  covado,  -  = 

24f  pollegada  =  1  covado  avantejado,        -        = 

13 J  pollegada  =  1  terca,  3  t.  =1  vara,  -  = 

Oporto ;  &  palma  =  1  covado,  -  = 

PRUSSIA  (legal  since  1820)  : 

12  zolle=  1  fus  (Rhein-fus),    -  -  -  = 

10  zolle  a  1  land-fus,  10  land-fus  or  12  Rhine- 

fus  =  1  ruth,  2000  r.  =  1  meile,  -        = 

25£  zolle  (Rhein-zolle)  =  lelle,  -  -  = 

180  square  ruthe  =  1  morgen. 

Dantzic  (special)  :  75  am  =  1  seil,  -  = 

Konigsberg ;  1  elle,  -  -  -  -  = 

RUSSIA  (legal for  the  Empire) : 

16  verschok  =  1  archine,   -  -  = 

3  archines  or  7  feet  =*  1  sachine,         -  -  = 

500  sachines  =  1  verst,       -  -  = 

2400  square  sachine  =  1  deciatine,      -  -  = 

Crimea. —  Sevastopol,  dfc. :  1  halebi,   -  = 

SARDINIA  I.  —  12  oncia  =  1  palma,       -  -  == 

22  oncia  =  1  pie,  = 

25^  oncia  =  1  raso,      -  -  -  -  = 

12  palmi  =  1  trabucco.     10  palmi  =  1  canna,  = 

SICILY  I.  —  Messina :  8  palmi  =  1  canna,     -        = 

Palermo  :  8  palmi  =  1  canna,     -  -  -  = 

SPAIN.  —  Alicant :  9  pulgada  =  1  palmo,  4  palmi 

=  1  vara,  2  vara  =  1  hraza,       -  = 

12  pulgada  or  lj  palmi  =  I  pie,  -  -  = 

Barcelona :  4  palmi  =  1  vara  or  matja-cana,         = 

2  vara  =  1  cana,  = 

Cadiz  (Standard  of  Castile)  :  —  6  nulgada  =  l 

sesma,  2  s.  =  1  pie  or  tercia,  3  pie  =  1  vara,  = 

5  pie  =3 1  passo. 

2  octava  =  1  quarta  or  palma,  4  q.  =  1  vara,   = 
12  ]»ulgada=l  pic,  1£  i»ie  =  l  codo,  2  c.  =  1 
vara,  2  v.  =  1   estado,   braza,    brazada  or 
toesa,  2  e.  =  1  cuurda,  2  c.  =»  1  cordel,  500 
cordcle=*l  legua,    -  -  -  -■■ 


U.  Stales. 
2.351    feet. 
0.557  yard. 

7.214    feet. 
1.279  miles. 

0.722 
0.744 
1.203 
0.707 

yard. 

»<. 

ft 

u 

1.03 

feet. 

4.68    miles. 
0.7293  yard. 

47.062 
0.628 

M 
M 

0.777 

7.— 

0.663 

2.7 

0.799 

0.286 

1.571 

feet. 

mile. 

acres. 

yard. 

a. 

feet. 

0.6 

2.87 

2.311 

2.07 

yard. 

M 

l.r.77 
0.833 
0.849 
5.094 

foot. 

yard. 

feet 

2.782 

u 

0.928 

yard 

4.215  miles. 

FOREIGN   LINEAE   AND   SURFACE   MEASURES.  U  21 

Foreign.  U.  States, 

192  vara  cuadrada  =  1  quartillo,  4  q.  =  1  ce- 
lemin,  7£  c.  =  1  arancada,  1§  a.  =  1  fane- 
gada,     -----        =    1.587  acres; 
50  fanegada  =  1  vugada. 
Corunna,  Ferrol :  4  palma  op  8  octava  wm  1  vara,  =    0.928  yard. 
Gibraltar.  —  As  at  Caaiz ;  also,  as  in  Great  Britain. 
Malaga.  —  Same  as  Cadiz . 

Santander :  8  octava  or  4  palma  =  1  vara,  -  =     0.913      " 

Valencia  :  9  onze=  1  palmo,  l£  p.  =  1  pie,  =     0.992    feet. 

3  pie  =  1  vara,  2\  v.  =  1  braza  reale,        -        =    2.232  yards. 
SWEDEN.  — 12  tuin  =  l  fot,  2  f.  =  lain,  -=    0.648     '* 

3  aln  =  1  Famn,  2§  famn  =  1  stang,  -        =  15.553    feet. 
2250  stang=l  mil,     -            -            -  -=    6.627  miles. 

4  kappland  =  1  fjerding,  4  f.  .=  1  spannland,  2 

s.  =  1  tunnland  =  218i|  square  stang,     -        =     1.211  acres. 
SWITZERLAND.  —  Legal,   since    1823,   for    the 
Cantons  of  Aarau,  Basle,  Betne,  Freiburg, 
Lucerne,  Solothurn,    Vaud ;  but  not  in  gen- 
eral use  : 
10  zoll  =  1  fuss,  4  f .  =  1  stab  mm  1  aune  of 

France,         -  -  -  -  -=  1.3124  yards. 

2k  stab  =  1  toise  or  ruthe,  -  -        =3.2809      " 

Special  and  local  — 

Basle :  12  zoll  =  1  schuh  or  fuss,  10  s.  *  1  ruthe,  =  3.33  " 

21A  zoll  =  1  braccio,       -  -  -  =0.5966      " 

44£    »    =  1  elle,  =  1.2348      " 

40^    "    =1  klafter. 
Berne :  12  zoll  =  1  fuss,  6  f.  =  1  klafter,  If  k.  or 

10  fuss=l  ruthe,     -  -  -  -=9.6215     feet. 

UJ  fuss  =  l  elle,  -  -  -  -        =  0.5933  yard. 

Geneva  :  12  zoll  =  1  pied,  5  J  p.=  1  toise,  -  =  8.528       feet. 

1  aune  (wholesale),       -  -  -  -=  1.299    yards. 

1  aune  (retail),      -  -  -  =  1.25         " 

Lausanne :    3f    piede  =  1    aune    (metrical,    of 

France),       -  -  -  -  -=  1.3123      " 

9  ])iL'de  =  1  ruthe. 
Neufchatcl ;  12  zoll  =  1  fuss,  10  f.  =  1  tois,     -      = 
22|  zoll  =  1  elle,  -  -  -  -  = 

St.  Gall :  10  zoll=  1  fus,  4  f .  =  1  stab,         -        = 
Zurich  :  12  zoll  =  1  fus,  2  f .  =  1  elle,    -  -  = 

a  elle  =  l  ruthe. 
TRIPOLI  (N.  Africa)  :  8  robi=  1  pic,  -        = 

1  pic  for  ribbons,  -  -  -  -  = 

TUNIS.  —  8  robi  =  1  pic,  for  woollens,  -        = 

1  pic,  for  silks,  -  -  -  -  = 

1  pic,  for  linens,   -  -  = 


9.621 

feet. 

0.608 

yard. 

1.3123 

<( 

0.6561 

ii 

0.6Q41 

tt 

0.5285 

(< 

0.736 

a 

0.690 

M 

0.5173 

a 

22  a  FOREIGN    LINEAR   AND   SURFACE  MEASURES. 

Foreign.  U.  States. 

TURKEY.  —  Aleppo :  8  robi  —  1  pic,        -  -  =  0.7396  yard. 

1  dra  mesrour,       ----=»  0.6089      " 

1  dra  stambouli,  -  -  -  -  =  0.7079      " 

Bagdad ;  1  guz,        -  ==0.8796      " 

Bussorah:  lguz=*  2.6389  ft.  l,hadid,  -=0.9502      " 

Constantinople :  1  halebi,      -  -  =  2.325      feet. 

1  endrasi  or  archim,     -  -  -  -  =  2.255         u 

1  pic  stambouli,  =  0.7079  yard. 

1  pic  for  silks,  -  -  -  -  =  0.7317      " 

Damascus :  1  pic,    -  =•  0.637        " 

Smyrna :  1  indise,  -  -  -  -  =  0.6846      " 

8  rob  =  1  pic,        -  =0.7302     " 

West  Indies. 

In  the  islands  of  Antigua,  the  Bahamas,  Barbadoes, 

Barbuda,  Dominica,  Grenada,  Jamaica,  Les 

Saints,  Montserrat,  Nevis,  Santa  Cruz,  St. 

John,  St.  Kitts\  St.  Thomas,  St.    Vincent, 

Tobago,  Tortola,  the  Measures  of  Length  are 

the  same  as  in  Great  Britain. 
In  Deseade,  Guadeloupe,  Mariegalante,  Martinique, 

St.  Lucia  : 
12  ponce  =  1  pied  de  Roy,  3§  p.  =  1  aune,        -  =»  1.3       yard* 
This  being  the  old  system  of  France,  or  system 

previous  to  1812. 
In  Bonaire,  Saba,  St.  Eustatius,  St.  Martin,  Same 

as  Holland. 
In  St.  Bartholomew,  Same  as  Sweden. 
In  Curacoa,  Trinidad,  Same  as  Castile  (Spain). 
Cuba  I.  —  Cardenas,  Cienfuegos,  Havana,  Matan- 

zas,  Nuevitas,  Porto  Principe,  St.  Jago,  &c.  ; 

Same  as  Castile  (Spain). 
Hatti  I.  —  Aux  Cayes,  Cape  Haytien,  Jeremie,  Port 

au  Prince,  Port  Platte,  &c.  :  Same  as  France, 

before  1812. 
Savanna,  <Sf.  Dmingo,  &c.  :    Same  as  Castile 

(Spain). 
Porto  Rico  I.  —  Ponce,  St.  Johns,  &c.  :  Same  as 

Castile  (Spain). 


23  a 


FOREIGN  WEIGHTS  REDUCED  TO  UNITED  STATES. 


Foreign.  U.  States. 

Avoirdupois 
pounds. 

ABYSSINIA.  —  Massuah  :    10  dirhem  =  1  wakea,  12 

w.  or  10  mocha  =  1  rotl  or  liter,  -  -  =      0.688 

ALGIERS.  —8  mitkal  —  1  wakea,  10  wakea  =»  1  rotl 

attari  {for  spices  and  drugs) ,  -  -        =       1.190 

18  wakea  =  1  rotl  gheddari  (for  fruits,  oil,  &c.),     =       1.339 
27  wakea  =  1  rotl  khebir  {market  pound),  -  =       2.008 

100  rotl  =  1  cantaro  or  quantar. 

1000  grammes  =  1  kilogramme,  -  =       2.205 

ARABIA.  —  Hodeida :  30  vakia  =  1  maon,  10  maon  = 

1  frazil,  40  frazils  =  1  bahar,      -  -  -  =  813. 

Jidda:  15  vakia  =  1  rotolo,  5  rotoli  =  1  maund,  =       1.83 

10  maunds  =  1  frazil,  10  f.  =  1  bahar,  -        =  183. 

Mocha:  15  vakia  or  wakega  =  1  rotolo  or  rotl,  -  =      1.5 

2  rotolo  =  1  maund  or  maon,  10  maunds  =  1  frazil, 

15  frazils  =  1  bahar,        -  -  -  -  =  450. 

100  miscals  =  1  vakia,  22£  v.  =  1  maund  copra,      =      2.25 
At  the  Bazaar,  for  coffee  — 

14£  vakia  =  1  rotolo,  and  a  bahar,       -  =  435. 

Muscat :  24  cucha  =  1  maund,  -  -  -  =       8.75 

AUSTRIA.  —  Trieste,   Vienna:  2  lothe  =  1  unze,  4  u. 
=  1  vierding,  2  v.  =  1  mark,  2  m.  =  1  pfund,  20 
p.  =  1  stein,  5  s.  =  1  centner,  -  =  123.47 

4  centner  =  1  karch.     1  saum,       -  -  -=339.5 

1  saum  for  steel,  -  -  -  =  308.666 

Trieste  (Venice  weight)  :   1  pfund  (peso  grosso) ,         -  =       1.052 
1  pfund  (peso  sottile)  — apothecaries'  weight,      -        =      0.665 
Bohemia.  — 100  pfund  =  1  centner,       -  -  -=  113.4 

Prague:  16  unze  =  1  pfund,        -  -  =       1.26 

18  pfund  =  1  stein,  6  s.  =  1  centner,         -  -  =  136.08 

Hungary.  —  1  oka,  -  -  -  -  -        =      2.78 

AZORE  ISLANDS.  —  Same  as  Lisbon  (Portugal). 
BALEARIC    ISLANDS.  —  Majorca.  —  25  rotoli  =  1 

arroba,     -  -  -  -  -  -=    22.37 

4  arroba  =  1  quintal,  112  rotoli  =  1  oder,      -        =  100.217 
100  rotoli  barbaresco  =  1  quintal,  -  -  -  —    92.794 


24  a  FOREIGN   WEIGHTS   REDUCED   TO   UNITED    STATES. 

Foreign.  U.  States. 

Avoirdupois 
pounds. 

Minorca.  — 100  libra  =  1  cantaro,        -  -  -=    88.2 

100  rotoli  barbaresco  =  1  quintal,  .     -        =    81.727 

3  quintals  =  1  carga,  -  -  -  -  =  275.18 

BELGIUM.  —  Same  as  Holland. 
Previous  to  1820  — 
8  pond  =  1  stein,  12£  s.  =  1  centner,  -  =  103.659 

3  centner  =  1  schippond,  4  centner  =  1  charge. 

Waeg  of  coals,         -  -  -  -=  150. 

BERMUDAS  I.  —  Same  as  Great  Britain. 

BOURBON  I.  — 100  livres=l  quintal,  3  q.  =  l  charge,  =  323.765 

BRAZIL.  — 16  onca  =  1  arratel,  32  arratel  =  1  arroba,  =     32.501 

4  arrobas  =  1  quintal,  13£  q.  =  1  tonelada. 

Central  and  South  America. 
Balize,  Campeche,  Guatimala,  Honduras,  Laguna,  Leon, 

Nicaragua,  San  Juan,  San  Salvador,  Sisal,  <3fC. 
Buenos  Ayres,  Callao,  Carlhagena,  Coquimbo,  Guaya- 
quil, Laguayra,  Lima,  Maracaybo,  Montevideo,  Rio 
Hacha,  Truxillo,  Valparaiso,  &c. : 
16  onza  =  1  libra,  25  1.  =  1  arroba,  4  a.  =  1  quin- 
tal,   =  101.546 

Berbice,   Demcrara,   Essequibo,    Surinam.  —  Same   as 

Holland. 
Cayenne.  —  Same  as  France. 
CANARY    [SLA  N I  >S.  —  Same  as  Castile  (Spain) . 
OANDIA    I.  — 44  oka  =  1  cantaro,  -  -        =116.568 

CAPE   COLONY.  —  ( tope  Town.  —  32  loot  =  1  pond,  =      1.03 
CAPE  VER  DE  ISLANDS.—  Same  as  Lisbon  ( Portugal) . 
CHINA.  — 10  lis  =  1  tael  or  Leung,  16  taels  =  1  catty 

$T  kan,  LOO  catties  =  1  pecul  or  tarn,      -  -=  133.333 

■2-2\  oka  =  1  leang,  10  1.  =  1  catty,  2  c.  =  1  yin, 

15  y.  =  1  kwan,  3£  k.  =  1  tarn,  1£  t.  =  1  shik,  =  160. 

CORSICA  I.  —  1  Itae,  -  -  -  -  -  =      0.76 

CYPRl'S  I.  —  4<)0<lrachmi  =  loka,40  oka=lmoosa,  =  112. 

l$oka=]  rotolo,  =       5.25 

DENMARK.  —  32  lod  =  16  unze  =  2  mark  =  1  pund,  =       1.101 
I  no  pound  =  1  centner,       ...  -=110.11 

12  pond  =  1  biamertmnd,  lj  b.  =  1  lispund,  20  1. 
=  1  drippund,     -  -  -  -  -=  862.364 

2|  lisjumd  ■■  1  waag,  -  -  =     39.639 

EGYPT.  —  144  drachm i  or  100  miscali  =  1  rotolo  for- 

fora, =      0.95 

■ldti  dnchmi  ■■  1  harsela,  -  -  -        =      2.639 

420  drachmi  =  1  oka,         -  -  -  -  =      2.771 


FOREIGN   WEiailTS  REDUCED  TO   UNITED  STATES.  O  29 

Foreign.  ,  U.  States. 

Avoirdupota 
pounds. 

100  rotoli  or  36  harseli  =  1  cantaro  forfora,  -  =    95. 

105  "  or  36  oka  =  1  cantaro,  for  coffee,  -  =  99.75 
1  rotolo  mina,  for  spices,     -            -            -  -==       1.403 

1     "      zaidino,  for  dye-woods,  -  =       1.138 

1     "      zauro,  for  iron,      -  -  -  -=       2.216 

102  rotoli  =  1  cantaro,  for  quicksilver  and  vermilion,  =    96.9 

115  "  =1  "  for  almonds  and  fruit,  -  =109.25 
125  "  =1  "  for  drugs,  -  -  -=118.75 
133  "  =1  "  for  gum  arabic,  -  =126.35 
150     "     =1       "        for  plumbago,           -            -=142.5 

FRANCE.  — 1000  milligrammes  =  100  centigrammes  = 
10   decigrammes  =  1   gramme  =  15.43315   troy 
grains. 
100  grammes  =  10  decagrammes  =  1  hectogramme,  =    .  0.22 
10  hectogrammes  =  1  kilogramme,  -  -  =      2.205 

10  kilogrammes  =  1  myriagramme,      -  -        =     22.047 

100  myriagrammes  =  10  quintal  =  1  tonneau. 
1  livre  poids  de  marc,  -  =1.07922 

1  livre  metrique  =  £  kilogramme,         -  =  1.10237 

GERMANY.  —  Baden.—  Heidelberg,  Manheim,  &c.  : 
1000  as  =  100  pfennig  =  10  centas  =  1  zehnling. 
1000  z.  =  100  pfund  =  10  stein  =  1  centner,  „     -  =  110.237 
8  quentchen  =  2  loth  =  1  unze. 

16  unze  =  2  mark  =  1  pfund,  -  =       1.102 

Bavaria.  —  20  pfund  {legal)  =  1  stein,  -  -  =    24.693 

Augsburg  :  512  pfennig  =  128  quentchen  =  32  loth 

=  16  unze  =  2  mark  =  1  pfund  or  frohngewicht,  =       1.082 
100  pfund  =  1  centner,       -  -  -  -  =  108.262 

3  centner  =  1  pfundschwer  or  schiffpfund,       -        =  324.786 
7£  pfundschwer  or  100  stein  =  1  tonne. 

22£  pfund  =  1  stein,  5j  stein  =  1  wage,    -  -  =  129.914 

Nuremberg :  Denominations  and  relative  value,  same 
as  Augsburg. 
100  pfund  =  1  centner,  =  112.432 

Hanover  (legal)  :  32  ortchen  =  16  drachma  or  quent- 
chen =  2  loth  =  1  unze,  8u.  =  l  mark,  2  m.  = 

1  pfund, =      1.079 

14  pfund  =  1  liespfund,  -  -  -        =15.11 

24  liespfund  =  1  pfundschwer  or  last,        -  -  =  362.648 

20  pfund  =  1  stein  for  flax,  =    21.586 

Bremen :  32  ortchen  =  8  quentchen  =  2  loth  =»  1  unze, 

8  u.  =  1  mark,  2  m.  =  1  pfund,       -  -        =      1.099 

116  pfund  =  1  centner,       -  -  -  -  =  127.516 
2£  centner  —  1  schiffpfund,  =  318.791 

0 


26(1  FOREIGN  WEIGHTS  REDUCED  TO  UNITED  STATES. 

Foreign.  U.  Stales. 

Avoirdupois 
pounds. 

300  pfund  =  1  pfundschwer  or  last,  -  -  =  329.784 

14  pfund  =  1  liespfund.     1  wage,  for  iron,      -        =  131.913 
1  stein,  for  fax  =  21.985  lbs.     1  stein,  for  wool,      =     10.993 
Ernden,  Osnaburg :  10  unze  =  2  mark  =  1  pfund,  100 

pfund  =  1  centner,  -  -  -  -  =  109  J041 

3  centner  =  1  pfundschwer. 
Hesse    Cassel.  — 16  unze  =  1  pfund,  100  pfund  =  1 

centner,  -----=  106.762 

21  pfund  =  1  kleuder. 

Hesse  Darmstadt  {legal)  :  100  pfund  =  1  centner,  =  110.236 

Frankfort :  16  unze  or  2  mark  =  1  pfund,      -  -  =       1.114 

I          100  pfund  =  1  centner,             -            -            -  =111.407 

22  pfund  =  1  stein,             -            -            -  -  =    24.509 
HoLSTfiiN.  —  Hamburg,  Lubcc,  Altona,  Kiel: 

16  unze  =  2  mark  =  1  pfund,        -  -  -  =       1.068 

14  pfund  =  1  liespfund,  8  1.  =  1  centner,         -        =  119.6 

2£  centner  =  1  schiffpfund,  -  -  -  =  299. 

7f  schiffpfund  or  100  stein  =  1  tonne,  -        =  2135.72 

90  pfund  =  1  last,  -  -  -  -=     96.11 

1  stein ,  for  feathers  and  ivool,    -  -  =     10  679 

.  320  pfunds  {land freight)  =  1  schiffpfund,  -  =  341.72 

Mecklenburg    ^generally)  :     16   unze  =  2   mark  =  1 
pfund,  15  pfund  =  1  liespfund,  20  1.  =  1  schiff- 
pfund, ....-=  313.723 
Rostock:  16  pfund  =  1  liespfund,  20  1.=  1  s.-hi  tip  fund,  «  868  64 
280  pfund  =  1  schiffpfund, /or  iron  and  had.  -  =  313.723 
22  pfund  =  1  stein,  y  or  wool  and  fax,  -             -        =24.65 
Saxony.  —  Dresden,  Lcipsic:  32  pfennig  =  8  drachma  or 
quentlein  =  2  loth  =  1  unze,  8  u.  =  1  mark,  2 
mark  =1  pfund,              -            -            -             -=       1.03 
22  pfund  —  1  stein,  2  s.  =  1  wage,       -            -        —     46.324 
110  pfund  =  1  centner,       -             -             -             -=113.31 
114     M     «■  1      "      berg-gewfcht. 
118     "     =1      "       Btafl-gewioht 
Fni/hirg:  1  pfund,            ...  ==       1.165 
Oldenburg.  —  Denominations  and  relative  values,  same 

as  Bremen,  but  weight  vahu «  li.'.>4 %  less. 
GREAT  BRITAIN.*  —  Bee  United  States. 

*  In  Gnat    Britain,  in  addition    to  the  denominations  of  weights  used  in  the  United 
States  (the  values  of  which  Bit  tin-  MMM),  1 1  i . - 

Stone  oi  Inii'-h-r.s' ni.at  or  lli-.-h,   =      8  lbs. 

St ofcheoMj =   16   ** 

:   'lass, B 

=  ISO    •• 

BtootofMnm, m 

Fotbxr  of  lead,   ....    •  .   .   .  =  19i  cwt. 


of  WOOl, 

=        7  lbs 

'       "     iron,  flour, 

=      14    - 

Tod       «       " 

=      i 

Weigh "     " 

=    182    " 

Sack    "      " 

=    864    « 

L»gt     «       »t 

b=4368    " 

FOREIGN    WEIGHTS   REDUCED   TO   UNITED   STATES.  a  27 

Foreign.  U.  States. 

▲tolrdapoii 
ptwnflfi 

GREECE.  —Athens :  400  draclimi  =  1  oka,     -  -  =      3.137 

44  oka  =  1  eantaro,      -  =  148.3 

Morea.  —  1  eantaro,  generally,  -  -  -  -■■123.75 

Patras  :  400  draclimi  =  1  oka,     -  -  =       2. 043 

44  oka  =  1  eantaro  or  quintal,        -  -  -=110.3 

HOLLAND  (legal)  :    10"  korrel  =  1  wigtjc,    10  w.  =  1 

lood,  10  1.  =  1  onz,  10  o.  =  1  pond  =  1  kilogramme 

of  France,  -  -  -  -  -=      2.204 

2000  ponds  =  1  vat  (shipping),  -  -        =2204.74 

Previous  to  1820  — 

300  ponds  =  1  schippond,  -  -  -  -  =  32G.77 

8  ponds  =  1  steen,  =       8.71 

India  and  Malaysia  or  East  Indies. 

An-nam.  —  Cochin  China —  Saigon  :  16  luong  =  1  can, 

10  can  =  1  yen,  5.y.  =  1  binh,      -  -  -  =     68.876 

2  binh  =  1  ta,  5  ta  =  1  quan,  -  -        =  688.76 

Tonquin  —  Kesho  :  100  catties  =  1  pecul,       -  -=  132. 

Birmah.  —  Pegu :  12^  tical  =  1  abucco,  2  a.  =  1  agito, 

4  agiti  =  1  vis,  -  -  -  =      3.393 

33  tical  =  1  catty,  3  c.  =  1  vis,      -  -  -  =      3.393 

Rangoon :  2  small  rwes  =  1  large  nve,  4  large  rwes  = 
1  bai,  2  b.  =  1  mu,  2  m.  =  1  mat'h,  4  niat'hs 
=  1  kyat  or  ticul,  100  k.  =  1  paitktlia  or  vis,      =      3.65 
Borneo  I.  — 100  catty  =  1  pecul,  -  -  -==135.633 

Ceylon  I.  —  Colombo  :  500  pond  =  1  bahar  or  candy,     =  500. 

Celebes  I. — Macassar:  100  catty  =  1  pecul,        -        =  135.633 
Hindostan  —  Bengal,  generally  (bazar  weight)  : 

10  mace  =  1  khanelia,  3  k.  =  1  chattac,  10  c.  =  1 
dhurra  or  pussaree,  8  d.  =  1  maon  or  maund,       =     82.133 
Calcutta   (factory  weight)  :    5  sicca  =  1  chattac,  16  c. 

=  1  seer,  40  s.  =  1  maund,  -  -  -  =     74.666 

Bombay :  36  tanks  or  15  pice  =  1  tipprce,   2  t.  =  1 

seer,  40  s.  =  1  maund,  20  m.  =  1  candy,      -         =  560. 

2±  tank-seers  =  1  rupee-seer,  for  liquids  =  1.54  lbs. 

Goa  :  32  seers  =  1  maund,  20  in.  =  1  bahar,  -  =  495. 

Madras:  10  pagodas  or  varahuns  =  1  pollam,  8  p.  = 
1  seer,  5  s.  =  1  vis  or  visay,  8  v.  =  1  maund  or 

maon,       -  -  -  -  -  -  <=     25. 

20  maunds  =  1  candy  or  baruay.  -  =  500. 

Malabar  Coast:  40  polams  =  1  vis,  8  v.  =  1  maon,     =     30. 

20  m.  =  1  candy,  20  c.  =  1  garce. 
Malabar  (interior)  :  20  maon  =  1  candy,        -  -  =  095. 


28  a  FOREIGN  WEIGHTS  REDUCED  TO  UNITED  STATES. 

Foreign.  {J.  States. 

Avoirdupois 
pounds. 

Mangalore :  6  sida=  1  vis,  8  v.  =  1  maund,  20  maunds 

•    =  1  candy, =  564.72 

Massulipatam :    1£  nawtauk  =  1  chittac,    6  c.  =  1 
yabbolain,  2  y.  =  1  puddalum,  2£  p.  =  1  vis,  6| 

v.  =  maund,  20  m.  =  1  candy,  -  -  -  =  500. 

21j  vis  =  1  pucca  maund,        -  -  -        =    80. 

Mysore,  Seringapatam :    10  varahuns  —  1  pollani,  40 

p.  =  1  pussaree,  8  pussaree=  1  maon  or  maund,  =    24.276 
20  maund  =  1  bahar  or  candy. 
Pondicherry :  10  varahuns  =  1  poloin,  40  p.  =  1  vis, 

8  v.  =  1  maund,  20  m.  =  1  candy,         -  -  =  588 

Serampore,  Tranquebar  {legal)  :     Same  as  Denmark. 
Sinde:  2  pice  =  1  anna,  2  a.  =  1  chittac,  16  c.  =  1 

seer,  40  seers  =  1  maund,  -  -  -  =  74.666 

Sural  (new  measure)  :    8  pice  =  1  tippree,  2  t.  =  1 

seer,  40  s.  =  1  maund,  21  m.  =  1  candy,     -        =  300. 

3  candi  =  1  bhaur. 
Tatta :  4  pice  =  1  anna,   16  a.  =  1  seer,  40  s.  =  1 

maund,    -  -  -  -  -  -  =    74.32 

Java  I.  —  Batavia :  16  tael  =  1  catty,  l£  c.  =  1  goelak, 
66|  g.  or  100  catty  =  1  pecul,  3  p.  =  1  bahar  = 
16  m.  1  vis,  24  pollams  of  Madras,  -  -  =  405.333 

4£  pecul  =  1  great  bahar. 

5  pecul  =  1  timbang,  for  grain,  -  =  675.555 

Bantam :    24  tael  =  1  goelak,  100  g.  =  1  catty,   2 

catty  =  1  bahar,  for  pepper,        -            -            -  =  405.333 
Malacca.  —  Malacca:  16  tael  =  1  catty,  100  c.  =  1  pe- 
cul, 3  peculs=  1  bahar,        -  -  -        =405. 

2£  pinga  =  1  tampang,  2  t.  =  1  bedoor,  12  b.  =  1 

hali,  14  h.  =  1  kip,ybr  tin,         -  -  -  =    40.677 

2  buncals  =  1  catty,  for  gold  and  silver,  -        =      2.049 

Philippine  I.  —  Manilla,  &c. : 

22  piastres  =  1  catty,  100  c.  =  1  pecul,  -        =  139.443 

1  caban  of  rice  (usual) ,       -  -  -  -=  133. 

1  caban  of  cocoa,  =    83.5 

Siam.  —  Bangkok,  &c.  :  4  tical  =  1  tael,  2  t.  =  1  catty, 

100  catty  =  1  pecul,       -  -  -  -  =  135.238 

Singapore  I.  — Same  as  Malacca. 
Sooloo  Islands.  — 10  mace  =  1  tael,  16  t.  =  1  catty, 

50  c.  =  1  lachsa,  2  1.  =  1  pecul,      -  -        =  133.333 

Sumatra  I.  — 62  catties  =  1  pecul,       -  -  -=132.587 

24  tael  =  1  salup,  2  s.  =  1  ootan,  7J  o.  =  1  nelli, 
for  camphor,  -  -  -  -  =*    29.333 


Avoirdupois 
pounds. 


FOREIGN   WEIGHTS  REDUCED  TO   UNITED   STATES.  a  29 

Foreign.  U.  States. 

Achecn :  10  mace  =  1  tael,  20  t.  =  1  goclak,  1£  g.  = 

1  catty,  36  c.  =  1  maund,  -  -  -  =    76.986 

5£  inaund  =  1  candil  or  bahar. 
Bencoolen:  46  catties  =  1  maund,  -  =     98.371 

5§  maunds  =  1  bahar,         -  -  -  -  =  557.416 

IONIAN    ISLANDS.  —  Cephalonia,     Corfu,     Ithaca, 
Paxos,  Zante,  &c. : 

Legal  since  1817  :  100  libbra  =  1  talento,  -        =  100. 

100  oke  =  385  marcs. 

44  oke  =  1  cantaro,  -  -  -  -  =  118.807 

Cephalonia:  04  libbra  =  1  barile,/or  salt,  -        =    67.262 

Corfu :  100  libbra  =  1  talento,  -  -  -=     90.034 

ITALY.  —  Carrara.  —  1    carrata,  for  marble,  =  25 

cubic  palma  =  12.764  cubic  feet  =  31f  centi- 

najo  or  29  J  quintale  of  Modena,        -  -        =2240. 

LOMBARDY   AND    VENICE. 

Government  and  Customs  Measure : 
10  grani  =  1  denaro,  10  d.  =  1  grosso,  10  g.  =  1 
oncia,  10  o.  =  1  libbra,  10  1.  =  1  rubbio,  10  r. 
=  1  centinajo  =  10  myriagrammes  of  France,        =  220.474 
Special  and  local : 

Venice.  —  Peso  grosso  :  4  grani  =  1  carato,  32  c.  =  1 
saggio,  6  s.  =  1  oncia,  6  o.  =  1  marco,  2  m.  = 

1  libbra, =       1.052 

25  libbre  =  1  miro,  40  m.  =  1  migliajo,     -  -  =1051.86 

Peso  sottile :  4  grani  =  1  carato,  24  c.  =  1  saggio, 

6  s.  =  1  oncia,  12  o.  =  1  libbra,       -  =      0.666 

100  libbre  =  1  quintale,  4  q.  =  1  carica,  -  -  =  266.332 

Naples.  —  20  acini  =  1  trapeso,  30  t.  =  1  oncia,  12  o. 

=  1  libbra,  26  1.  =  1  rubbio,  -  -        =    18.387 

150  libbra  =  1  cantaro  piccole,       -  -  -  =  106.08 

33J  onci  =  1  rotolo,  100  r.  =  1  cantaro  grosso,        =  196.45 
Sardinia. —  Genoa :  24 grani  =  1  denaro,  24  d.  =  1  oncia, 
12  o.  =  1  libbra,  l£  1.  =  1  rotolo,  16%  r. 
(25  libbre)  =  1  rubbio,  4  r.  =  1  centinajo,  l£  c.  = 

1  cantaro. 
Peso  grosso :  1  centinajo,  -  -  =     76.863 

Peso  scarso  :  1  centinajo,     -  -  -  -  =     69.875 

Nice :  12  oncia  =  1  libbra,  25  1.  =  1  rubbio,  4  rubbi 

=  1  centinajo,  :.-_==     68.694 

Turin:  25  libbre  =  1  rubbio,  -  -  -=    20.329 

States  of  the  Church.  —  1  libbra  Italiana,  -        =      2.204 

Ancona:  12  oncia  =  1  lira,  100  1.  =  1  cantaro,  -        =    72.942 

C* 


30  a  FOREIGN   WEIGHTS   REDUCED   TO   UNITED   STATES. 

Foreign.  U.  States, 

Avoidupois 

pounds. 

Rome:  12  oncia  =  1  libbra,  10  1.  =  1  decina,  10  d.  — 

1  cantaro,  -  -  -  -  -  =     74.763 

160  libbre  =  1  cantaro.     250  libbre  =  1  cantaro. 

1000  libbre  =  1  migliajo,       -  =  747.633 

Tuscan y.  —  Leghorn,  Florence,  Pisa : 

72  grani  or  3  donuri=  1  draimna,  96  draninie  or  12 

oncia  =  1  libbra,  100  1.  =  1  centinajo,   -  -  =     74.857 

10  centinaje  =  1  migliajo. 

160  libbre  =  1  cantaro  or  carara, for  wool,  fish,  <Sfc,  =  119.771 
50  rottoli  =  1  cantaro  generale  (old),         -  -  =  112.29 

JAPAN.  — 160  rin  =  10  pun  =  1  its-go    -         -      =     1.333  lbs. 
2,500  pun  =  1 00  ischo  =  1 0  itho  =  1  its  'ko-koo  ==      333£  lbs. 
MALTA  I.  —  30  trapesi  =  1  oncia,  30  o.  ==  1  rotl, 

100  rotl  =  1  cantaro  sottile,  =     174.504 

110  rotl  =  1  cantaro  grosso. 
MADEIRA  I.  —  Denominations    and    relative    values, 
.    same  as  Lisbon,  Portugal: 

32  arratel  or  libbra  =  1  arroba,      -  -  -  =     32.349 

MAI  Kill  US  I.  —  Port  Louis  :  16  onces  =  1  livre,         =       1.08 
MEXICO.  —  Standard  same  as  Cadiz,  Spain. 

MOROCCO.  — 100  rotl  ==  1  cantaro,  -  -        «  118.723 

Tangier s :  1  rotl  {miners'),     -  -  -  -=      1.06 

1  rotl  (market),    -  -  -  =      1.701 

MOZ  AMBlQ  U  E  (Africa) . —  Mozambique :  Same  as  Port- 
ugal. 
NORWAY.  —  Same  as  Denmark. 

PERSIA.1 — Bushire:  3  clieki  =  1  ratal,  7£  r.  1  maund 
tabree,  2  m.  tabree  =  1  maund  show. 
1  maund  show,  bazar,   -  =     12.5 

1      "       copra,     "  -  -  -  -=7.3 

1      "       show,  factory,  -  -  =     13.5 

1      "       copra,     "  -  -  -  -  =       7-75 

Tauris:    2  mascais  =  1  dirhem,    50  d.  =  1  ratel,    6 

ratcl  =  1  batman,  =       5. 017 

K/iiraz:   1  batman,       -  -  -  -  -=     10.123 

POBTUGAL. —  576  grao  or  24  escropulo  or  8  oatuava, 
=  1  onca,  16  o.  =  4  quarto  =  2  maroa  =  1  arra- 
tel,   -  -  -  -  -  =      1.016 
32  arratel  —  1  arroba,  4  arrobe  =  1  quintal,         -  =  130.06 
13^  quintale  =  1  fcoaelada. 
PR1  8SL4    {Irgal   stare  1820):    4   quentchen  =  1  loth, 

21.  — J  un/.e,  8  u.  =  1  mark,  2  m.  1  piund,         =     1.0312 
I64  pfund  =  1  li.spiund,  1|  1.  =  1  stein,  -  -  =     22.687 

5  stein  =  1  centner,  3  0,  =  1  schillplund,         -         =  340.31 


FOREIGN   WEIGHTS  REDUCED  TO   UNITED   STATES.  a  31 

Foreign.  U.  States. 

Avoirdupois 
pounds. 
100  pfunds  =  1  lagel,  for  steel,       -  -  -  =  103.116 

1  Prussian  mark  =  1  Cologne  mark. 

Dantzic:  33  pfund  =  1  Btein,  for  fax,      -  =     34.029 

RUSSIA  (bgal  throughout  the  Umpire)  : 

3  zolotnik  =  1  loth,  32  loth  or  12  lana  =  1  funt, 

40  font  mm  1  pud,  -  -  -  -  =    30.067 

10  pud  =  1  berkowitz,  3  b.  =  1  paken,  -        =1082.02 

2  paken  =  1  last. 

Libau:  20  funt  =  1  licspfunt,  20  1.  =  1  schiffpfunt,  =  364.168 

Hamburg  weights  arc  also  used  here. 
Riga :  20  pfunde  =  1  liespfund,  5  1.  =  1  lof,  -  =     92.158 

4  lof  =  1  berkowitz  or  gehiffpfund,       -  -        =  368.633 
SARDINIA    I.  —  Cagliari,   &c.  :  12  oncia  =  1  libbra, 

26  1.  =  1  rubbio,  4  r.  =  1  cantarello,     -            -  =  93.082 

SICILY  I.— Messina:  12  oncia  =  1  libbra,             -        =  0.707 

2£  libbre  =  1  rotolo  sottile,  -  -  -=       1.768 

33  oneia  =  1  rotolo  grosso,       -            -                     =  1.945 

100  rotoli  =  1  cantaro  (gross  or  net) . 

Palermo :  250  libbre  or  100  rotoli  sottile  =  1  cantaro 

sottile,      -            -            -            -                        -=  175.04 

275  libbre  or  100  rotoli  grosso  =  1  cantaro  grosso,  =  '192.556 
Syracuse :  250  libbre  or  100  rotoli  sottile  =  1  cantaro 

sottile,            -            -            -            -                     =  180.125 
275  libbre  or  100  rotoli  grosso  =  1  cantaro  grosso,  =  198.137 
SPAIN.  — 16  Castilian  onze  =  1  Castilian  libra  ( Cus- 
toms'),       =  1.01546 

Alicant:  12  onze  =  1  libra  menor  (minor). 

18  onze  =  1  libra  mayor  (major),          -                      =  1.144 

24  1.  mayor  or  36  1.  menor  =  1  arroba,      -            -  =  27.456 

4  arrobe  =  1  quintal,  2£  q.  =  1  carga,             -        =  274.567 
8  carga  =  1  tonelada. 

24  Castilian  libre  =  1  arroba,/or  vermilion,            -  =  24.371 

25  Castilian  libre  =  1  arroba  of  the  Customs,  -        =  25.386 
'Barcelona:  25  libre  =  1  arroba,          -            -            -=  22.14 
Bilboa :  25  libre  =  1  arroba,         -                                    =  26.97 
Cadiz  (Standard  of  Castile)  :    8   onza  =  1  marco, 

2  m.  =  1  libra,  25  1.  =  1  arroba,  4  a.  =  1  quintal,  =  101.546 
20  quintale  =  tonelada. 
Corunna,  Ferrol :  16  onze  =  1  libra  sutil,  100  libre 

sutil  =  l  quintal  (Castilian),        -  -  -=101.546 

20  onze  =  1  libra  gallega,  100  1.  g.  =  1  quintal,      =  126.933 

25  libre  =  1  arroba. 

Gibraltar :  16  onze  =  1  libra  (  Castilian) ,  -        =1.01546 

16  ounces  =  1  pound,  25  p.  =  1  arroba,   -  -  =    25. 


32  a 


FOREIGN   WEIGHTS  REDUCED  TO   UNITED  STATES. 


Foreign. 


U.  Slates. 

Avoirdupois 
pounds. 

=  177.7 
=  152.28 
=      0.784 
=      1.176 

-  =  338.413 


Malaga.  — Same  as  Cadiz. 

1|  quintal,  or  3£  barrile  =  1  carga  of  raisins, 
Santander:  100  libre  =  1  quintal, 
Valencia :  12  onze  =  1  libreta  or  libra  menor, 
18  onze  =  1  libra  gruesa,  - 

35   libre    menor  =  1    arroba,   4    a.  =  1    quintal, 
12j  arobe  =  1  carga,       - 
SWEDEN.  —  Stockholm,  &c.  : 

Viktualie-wigt  or  skal-wigt :  4  quintin  =  1  lod,  2  1. 

=  1  untz,  16  u.  =  1  skalpund,  -  -        =    0.9375 

20  skalpund  =  1  lispund,  20  1.  =  1  skeppund,       -  mm  375. 

32  skalpund  =  1  sten,  -  -  =     30. 

12  skeppund  =  1  last. 
Metall-wigt    or    jern-wigt    (for    iron,    steel,    &c.)  : 
20   mark  =  1  markpund,    20    m.  =  1    lispund, 

20  1.  =  1  skeppund,        -  -  -  -  =  300. 

15  skeppund  =  1  last. 

8£  lispund  =  1  waag  or  vog,  for  tin,  -  -  =  123.75 

Uppstadt-wigt  (inland  weight)  : 

400  pund  or  20  lispund  =  1  skeppund,       -  -  =  315.674 

Tachjern-wigt :  400  pund  or  20  liap.  =  l  skeppund,  =  453.47 
Berg-wigt :  400  pund  or  20  lisp.  =  1  skeppund,      =  348.822 
SWITZERLAND  (legal,  since  1823,  for  the  Cantons  of 
Aarau,  Basle,  Berne,  Freiburg,  Lucerne,  Solothum, 
Vaud ;  but  not  in  general  use)  : 
8  gros  =  1  unze,  8  u.  =  1  mark,  2  m.  =  1  livre  or 

pfund  =  1  livre  poids  de  marc  of  France,  =  1.07922 

10  livres  =  1  stein,  10  s.  =  1  centner,        -  -  =  107.922 

Special  and  local : 

Berne:  100  pfunde  =  1  centner,  -  =  114.9 

Geneva :  24  grani  =  1  denier,  24  d.  =  1  once,  15  o. 

=  1  livre  foible, =  1.0118 

18  once  —  1  livre  /brt,  -  -  -  -        =1.2141 

Lausanne :  16  onces  =  1  livre  =  1  livre  poids  de  metrique 

of  France,  -  -  -  -  -  =  1.10237 

Neufchatel:  8  onces  =  1  marc,  2  m.  =  1  livre,     •         =  1.14682 
St.  Gall  :  10  unze  =  1  loth,  2  1.  =  1  pfund,  -  =     1.2&14 

Zurich:  18  unze  =  1  pfund,  -  -  =     1.1637 

Poids  foible  (for  silks,  &c.)  :  2  lothe  =  1  unze,  8  u. 

=  1  mark,  2  m.  =  1  pfund,  -  -  -         =     1.0344 

TRIPOLI.  — (N.  Africa)  :  S  termini  =  1  usano,  16  u. 

=  1  rotolo,  100  rotoli  =  1  cantaro,        -  -  =  111.214 

400  drachmi  =  1  oke,  «    2.7429 


FOREIGN   WEIGHTS  REDUCED   TO   UNITED   STATES.  a  33 

Foreign.  U.  States, 

Avoirdupois 
pounds. 

TUNIS.  — 8  metical  =  1  usano,  16  u.  =  1  rotolo,  100 

rotoli  =  1  eantaro,    -  =  109.155 

TURKEY.  —  Aleppo :  266|  meticals  —  1  okc,   -  -  =  2.81349 

480  meticals  =  1  rotolo,  5  r.  =  1  vesno,  20  v.  =  1 

eantaro,  -       ■■  506.428 

3J  rotoli  =  1  batman,  10£  b.  —  1  cola,     -  -  =  177.249 

30£  rotoli  =  1  eantaro  zurlo,     -  -  =  154.46 

400  meticals  =  1  rotolo  for  Dainacene,        -  -  =  4.22023 

453J  meticals  =  1  rotolo  for  Persian  silks,        -        =  4.78293 
460^  meticals  =  1  rotolo  Tripolitan,  -  -  =  4.92361 

Bagdad:  2£  vakia  =  1  oke,  =  2.74286 

Bussorah:   J  00  miseals  =  1  cheko,      -  -  -=1.02857 

24  vakia  =  1  maund,     -  =  116. 

46  oke  =  1  cuttra,  -  -  -  -  —  136.482 

24  vakia  attaree  =  1  maund  attaree,    -  =     28. 

76  vakia  attaree  =  1  maund  sessee,  -  -  =  88.6666 

4f  vakia  attaree  =  1  vakia. 
Constantinople:  16   kara,   kilot  or  taim  =  1  dirhem, 
100  dirhem  or  06j  meticals  =  1  cheki  or  yusdrum, 
2  cheki  =  1  rottel,  100  r.  =  1  eantaro,  -        =  140.3 

13T7T  rottel  =  1  batman,  7£  b.  =  1  eantaro. 
266^  meticals  or  2  rottel  =  1  oka. 

1 16|  meticals  =  1  cheki,  for  opium,  -  -  =  1.7578 

Damascus :  400  meticals  or  60  peso  =  1  rotolo,  100 

rotoli  =  1  eantaro,     -  =  395.673 

Smyrna:  100  drachmi  or  66f   miseals  =  1  cheko,  2£ 

cheki  =  1  cequi,  -  -  -  -  -  =  1.7578 

180  drachmi  or  120  miseals  =  1  rotolo,  13J  r.  =  1 

batman,  7£  b.  or  100  rotoli  =  1  eantaro,       -        =  126.571 
4  cheki  =  1  oka,  45  o.  =  1  eantaro. 
44  oke  =  1  eantaro,  for  tin,  &c.,    -  -  -  =  123.758 

West  Indies. 

In  the  Islands  of  Antigua,  The  Bahamas,  Barbadoes, 
Barbuda,  Dominica,  Grenada,  Jamaica,  Les 
Saints,  Montserrat,  Nevis,  St.  Kitts,  St.  Vincent, 
Tobago,  Tortola,  the  commercial  weights  are  the 
same  as  in  Great  Britain. 

In  Deseade,  Guadeloupe,  Mariegalante,  Martinique,  St. 
Lucia  :  2  quartiers  =  1  marc,  2  m.  =  1  livre,  100 
1.  =  1  quintal,      -  -  -  -=107.922 

3  quintals  =  1  charge,  3£  c.  =  1  millier. 
This  being  the  old  system,  poids  de  marc,  of  France. 


34  a  FOREIGN  WEIGHTS  REDUCED  TO  UNITED  STATES. 

Foreign.  U.  States. 

Aroirdupois 
pounds. 

In  Saba,  St.  Eustatia,  St.  Martin,  the  commercial  weights 

are  the  same  as  in  Holland,  old  system. 
In  Santa  Cruz,  St.  John,  St.  Thomas,  Same  as  Denmark. 
In  St.  Bartholomew,  Same  as  Sweden. 
In  Curacoa,  Trinidad,  Same  as  Castile  (Spain). 
In  Bonaire:  100  pond  =  1  centenaar,    -  -  -  s  103.659 

3  centenaar  =  1  schippond. 
This  being  the  old  weight  of  Brabant,  Holland. 
Cuba  I. —  Cardenas,    Cienfuegos,    Havana,    Matanzas, 

Nuevitas,  Porto  Principe,  St.  Jago,  &c.  :  Same  as 

Castile  (Spain). 
Hayti  I.  —  Aux  Cayes,  Cape  Haytien,  Jeremie,  Port  au 

Prince,  Port  Platte,  &c.  :  Same  as  France,  before 

1812. 
Savanna,  St.  Domingo,  &c. :  Same  as  Castile  (Spam). 
Porto  Rico  I.  —  Ponce,  St.  Johns,  &c. :  Same  as  Castile. 


35  a 


FOREIGN  LIQUID   MEASURES  REDUCED  TO   UNI- 
TED STATES. 

Foreign.  U.  States 

Wine 
gallons 

ABYSSINIA.—  Massuah:  1  cuba,         -            -            -  m  0.268 
ALGIERS.  — 16|  litres  —  1  khoulle,  6  k.  =  1  hectoli- 
tre,           =  26.418 

ARABIA.  — Mocha:  20  vacias  {weight)  =  1  nusfiah,  8 

n.  =  1  cuda  or  gudda  =  10  lbs.  Av. ,  or  of  oil,  &c.  =  2.07 
AUSTRIA  {legal  and  general  for  the  Empire)  : 
Vienna,  Trieste,  Lintz,  Prague,  Pesth,  &c.  : 

4  seidel  —  1  mass,  10  mass  =  1  viertel,             -        =  3.738 

4  viertel  =  1  eimer  or  orna,             -            -            -  =  14.952 

32  eimer  =  1  fuder,    .  -            -            -                     =  478.48 
Special  and  local  — 

Trieste :  4|  caffiso  =  1  orna,                -            -            -  =  14.952 

Bohemia.  —  32  pinte  =  1  eimer,       -            -                     =  16.141 

4  eimer  =  1  fass,     -            -            -            -            -  =  64.56 
Prague:  60  mass  =  1  eimer,         -            -            -        =16.591 
Hungary.  =  4  r impel  =  1  halbe  or  icze,  100  icze  =  1 

czeber,     -            -            -            -            -            -  =  22. 

1  an  thai  =  13.352  gals.  1  ako,             -                     =  18.495 

Buda,  Pesth,  &c.  :  1  eimer,     -            -            -            -  =  15.028 

Presburg :  1  eimer,            -            -            -                     =  19.368 

Moravia.  —  40  mass  =  4  viertel  =»  1  eimer,       -            -  =  11.282 

AZORE  ISLANDS.  —  Same  as  Lisbon  {Portugal). 

BALEARIC  ISLES.—  Majorca.  — 4  quarta  =  1  quartes, 

6  quartes  =  1  cuartin,  4  cuartin  =  1  carga,          =  28.666 
3  carga  =  1  pelexo.     1  quartinello,      -            -        =1.8 
Minorca.  —  4  quarta  =  1  quartes,  3  quartes  =  1  gerrah, 

10  g.  =  1  carga,  4  c.  =  1  botta,             -            -  =  133.379 

1  barrel, =  8.344 

BELGIUM.  —  3 J  canette  =  1  uper,  10  u.  =  1  emmer, 

3  e.  =  1  vat  =  1  hectolitre  of  France,      -            -  =  26.418 


36.578 


2  pinte  =  1  pot  or  mingle,  2  p.  =  1  stoop  or  gelte, 

2  s.  =  1  schreef,  25  schreef =  1  aam,  -        = 

6£  aam  =  1  ton  of  spirits,  for  shipping,      -  -  =  237.76 

21  stoop  =  1  velte,  =      2.011 


36fl  FOREIGN   LIQUID  MEASURES  REDUCED   TO  UNITED   STATES. 

Foreign.  U.  States. 

Wine 
gallons. 

BERMUDAS  ISLANDS.  —Same  as  United  States. 

BRAZIL.  —  Bahia :  10  garrafa  =  1  canada,            -  =    18.734 

10  canada  =  1  pipa  of  molasses,     -            -  -  =  187.342 

7£  Canada  =  1  pipa  of  spirits,  -  -  =  134.885 
Rio  Janeiro  :  4  quartilho  =  1  medida,  3  m.  =  1  pote,  =       2.185 

16  pote  =  1  pipa,  2.  p.  =  1  tonelada,        -  -  =  262.178 

1  frasco,             -            -            -            -  =      0.562 

Central  and  South  America. 

Balize,  Guatimala,  Yucatan,  Bolivia,  Buenos  Ayres, 
Equador,  New  Granada,  Paraguay,  Peru,  Uru- 
guay, Venezuela.  Denominations  and  values, 
same  as  Castile  (Spain.) 

Guiana.  —  Berbice,  Demerara,  Essequibo,  Surinam :  Same 
as  Holland. 
Cayenne.  —  Same  as  France. 

CANARY  ISLANDS.  —  Grand  Canary,  Teneriffe,  &c.  : 

4  cuartilla  =  1  arroba,  28£  a.  =  1  pipa,  -        =  120.06 

CANDIA  I.— By  weight  —  1  oke  =  2.649  lbs.  Av. 

CAPE  COLONY.  —  Cape  Town:  16  flask  =  1  anker, 

4  anker  =  1  aain,  4  aam  =  1  legger,       -  -  =  152. 

CAPE   VERDE  I.  —  Same  as  Lisbon  (Portugal). 

CHILI.  —  Valparaiso,  Coquimbo,  &c.  :  4  copa  =  1  quar- 

tilla,  4  q.  =  1  arroba,  -  -  =       9.906 

CHINA.  —  By  weight.     See  Weights.     Also  — 

10  kop  tsong  =  1  shing  tsong,  10  shing  tsong  =  1 
tau  tsong,  5  tau  tsong  =  1  nok  tsong,  2  hok  tsong 
=  1  shik  tsong,  If  shik  tsong  =  1  yu,  5  yu  =  1 


ping  =  832  lbs.  Av. 
[CA  I. — 


CORSICA  I.  —  4  cuarto  =  1  boccale,  9  b.  =  1  zucca, 

12  z.  =  1  barile,  -  -  -  -  =     36.985 

CYPRUS  I.  —  By  weight.     See  Weights. 
J )  IA.M  ARK.  —  155  pagel  or  38|  potte  or  19|  kande  = 

1  anker,         -...-==      9.889 
4  anker  =  1   aam,   1£  a.  =  1  oxehoved,  4  o.  =  1 

fader,  1|  f.  =  1  stykfad,  -  -  -  =  296.672 

60  vi.rt.l  =  1  piba,  =  122.499 

17  vii-rti-1  =  1  toende,  for  beer,       -  -  -=    34.708 

EGYPT.  —  By  weight.     See  Weights. 
FRANCE.  —  1000  millilitres  —  100  centilitres  =  10  de- 
cilitre* =  1  litre,  =      0.264 
I001itren=  HI  decalitres  =1  hectoliter     -             -  =     26.418 
100  hectolitres  =  10  kilolitres  =  1  myrialitre. 


FOEEIGN   LIQUID  MEASURES  REDUCED  TO   UNITED  STATES.     «  37 

Foreign.  U.  State$, 

Wiuo 
gallons. 

GERMANY.  —  Baden  {legal)  :    4  schoppen  =  1  mass, 
12£  m.  =  1  stutz,  8  s.  =  1  ohm  =  1^  hectolitres  of 

l-Yance, m    39.626 

10  ohm  =  1  fuder. 
Manheim :  16  schoppen  or  4  eich-niass  =  1  viertel,  12 

vicrtel  =  1  ohm,  -  -  -  -  an    25.285 

16  schoppen  or  4  wirths-mass  =  1  viertel,  24  viertel 
=  1  ohm  =  16  decalitres  of  France,  -  =*    42.268 

Bavaria  {legal)  :  4  quartil  =  1  mas  or  masskanne,  60 

masskanne  =  1  eimer,      -  -  -  -  =     16.944 

|S4         M  =1  eimer,  for  beer,  -  =     18.075 

,     Augsburg,  Wurtzburg :  4  achtel  =  1  seidel,  16  s.  =  1 

boson,  8  beson  =  1  eimer,  12  e.  =  1  fuder,         -  =  238.745 
Nuremberg:    visir-niass  :    4  seidel  =  1  viertel,  32  v. 

=  1  eimer,  12  e.  =  1  fuder,  =  232.348 

Schenk-mass  :  32  viertel  =  1  eimer,  -  -  =     18.244 

12  eimer  =  1  fuder,  =  218.924 

Ratisbon :  32  viertel  =  1  eimer,  12  e.  =  1  fuder,         -  =  158.507 

Hanover  {legal) :  8  nossel  =  4  quartier  =  2  kanne  =  1 

stubchen,       -  -  -  -  =       1.036 

10  stubchen  =  5  viertel  =  1  anker,  -  -  =     10.36 
4  anker  or  2£  eimer  =  1  ahm,  -            -  =    41.439 
6  ahm  or  4  oxhoft  =  1  fuder,          -            -            -  =  248.637 
4  ahm  or  tonne,  for  beer,  =  1  fass,       -  =  165.756 

Bremen :  180  mingel  =  90  versel  '=  45  quartier   or 

,  vierling  =  ll£  stubchen  =  5  viertel  =  1  anker,    =      9.574 
24  anker  =  6  ohm  =  1  fuder,  -  -        =  229.779 

16  mingles  =  1  stubchen,   6  s.  =  1  stechkanne,  6 

stechkanne  =  1,  tonne,  for  whale  oil,        -  -=    30.64 

44  stubchen  {beer  measure)  =  1  tonne,  -         =    43.836 

Hesse  Cassel  :  4  schoppen  =  1  mass,  4  m.  =  1  viertel 

or  quartlein,  20  v.  =  1  ohm,  6  ohm  =  1  fuder,     =  251.547 
20  viertel  {beer  measure)  =  1  ohm,  -  -  =     46.128 

Hesse  Darmstadt  {legal)  :  Denominations  and  relative 
values,  same  as  H.  Cassel. 
1  ohm  =  16  decalitres  of  France,  -  =    42.27 

Frankfort:    4  schoppen  =  1  mass,  4£  neu-mass  =  1 

viertel,  20  v.  =  1  ohm,  6  ohm  =  1  fuder,  -  =  227.352 

8  alt-mass  =  9  neu-mass. 

11  fuder  =  1  stiickfass,  =  303.136 
Holstein.  —  Hamburg,  Altona,  Lubec :  8  oessel,  plank  or 

nossel  =  4  quartier  =  2  kanne  =  1  stubchen,  =      0.955 

8  stubchen  =  4  viertel  =  1  eimer,  -  -  -  =      7.644 

•  5  eimer  or  4  anker  =  1  ahm  or  fass,     -  -  =38.22 

D 


38  a     FOREIGN   LIQUID  MEASURES  REDUCED  TO   UNITED   STATES. 

Foreign.  77.  States. 

Wine 
gallons. 

l£  ahm  or  l£  tonne  =  1  oxhoft,      -            -            -  =  57. 33 

4  oxhoft  or  2  pipe  =  1  fuder,                                      =  229.32 
16  margel  or  melgel  =  1  stechkanne,  6  8.  =  1  tonne, 

2  t.  =  1  quarteel,/or  whale  oil,   -            -            -  =  61.548 

1  tonne,  for  beer,                                                            =  45.804 
Mecklenburg.  —  Rostock,  &c. :  Same  as  Holstein. 

Saxony.  —  Dresden :  8  quartier  or  2  nossel  =  1  kanne, 

quart  or  shenkkanne,  3  kanne  =  1  viertel,  -  =      0.743 

18  v.  =  1  anker,  lj  a.  =  1  eimer,         -  =     17.833 

2  e.  =  1  anker,  l£  a.  =  1  oxhoft,  -  -  -  =  53.499 
1|  o.  =  1  fass,  2f  f.  =  1  fuder,  -  -  =  213.996 
4  tonne  or  2  viertel  =  1  fass,  beer  measure,  -  -  =  104.026 

L'eipsic :  Denominations  and  relative  values  same  as 
Dresden,  but  capacity  values  =  12.53  °/0  greater. 
24  viertel  =  1  eimer,  =    20.067 

Freyburg :  100  schoppen  or  viertel  =  1  brente,  -  =     10.312 

16  brente  =  1  fass,  =  165. 

GREAT  BRITAIN :  Imperial  measure :  Denominations 

and  relative  values  same  as  wine  measure,  U.  S., 

but  capacity  values  20T3-rV  Per  cent«  greater.     See 

Liquid  Measures,  U.  S. 

GREECE.  —  Patras :  24  boccale  =  1  barile,      -  -  =    13.54 

HOLLAND  (legal)  :  10  vingerhoed  =  1  maatje,  10  m. 

=  1  kan,  10  k.  =  1  vat  =  1  hectolitre  of  F.,  =    26.418 

Previous  to  1820  — 
2  mutsje  =  1  pint,  2  p.  =  1  mingle,  2  m.  =  1  stoop, 
3Jj-  s.  =  1  viertel,  2$  v.  1  steekan,  2  s.  =  1  anker, 

4  a.  =  1  aam,  1£  aam  =  1  okshoofd,    -  =     61. 

1  aam,  for  oil,  -  -  -         *   -  -  =    37.64 

1  legger, /or  beer,  -  =153.57 

India  and  Malaysia  or  East  Indies. 

Ceylon  I.  —  Colombo :  4  aams  =  1  legger,         -  -  =  150.— 

IIi.mxotan.  —  Jiuinhay :  60  rupees  =1  seer,  50  seers  = 
1  inauiid  fluid  as  77  U»s.  Av. 
Calcutta:  5  sicca  =  1  chattac,  4  c.  =  1  pouah,  4  p. 
=  1  seer,   5  s.  =  1    BoaBaree,   8   p.  =  1   bazar 
maund  weight  =  82.18  lbs.  Av. 
Madras:  By  weight.     See  Weights. 
Seramporc,  JWwflwftflr,  {l<y<il)  \  Same  as  Denmark. 
Java  I.  —  Batana  :   ")  kau  =  1  barile,  -  -         =     13.207 

Luzon  I.  —  Manilla:  Same  as  Cadiz  (Spain). 


FOREIGN   LIQUID   MEASURES   REDUCED  TO    UNITED   STATES,   a  39 
Foreign.  U.  States. 

Wine 

gallons. 

Siam.  —  Bangkok :  20  canan  =  1  cohi  =  5  decalitres  of 

France,  -  -=     13.209 

Sumatra  I.  —  8  pakha  or  2  culah  =  1  koolah,  15  koolah 

—  1  tub, =    17.44 

IONIAN   ISLANDS.  —  Cephalonia :    2  quartucci  =  1 
boccale,  8  b.  =  1  pagliazza,  1£  p.  =  1  secchio,  G 

s.  =  1  barile,       -  -  -  -  -  =     18. 

Corfu,  Paxos :  l£  miltre  =  1  boccale,  12  b.  =  1  sec- 
chio, G  s.  =  1  barile,  -  -  =     18. 

Zante  :  1G  quartucci  or  8  boccale  =  1  lira,  1£  1.  mm  1 

secchio,  G  s.  =  1  barile,  -  -  -  -  =     18. 

ITALY.  —  Lombard?   and    Venice. —  Government  and 
Customs  measure:  10  coppi  =  1  pinta,  10  p.  =  1 
mina,  10  m.  =  1  soma  =  1  hectolitre  of  France,     »=     26.418 
Special  and  local  — 
Venice :  1£  quart  uzzi  =  1  boccale,  2g  b.  =  1  bozza,  4 

bozzi  =  1  secchio,  G  s.  =  1  inastello  or  concia       =     17.119 
2  niastclli  =  1  bigoncia,  4  b.  =  1  anfora,        -        =s  136.95 
14  anfori  =  1  botta,  -  -  -  -  =  171.187 

16  iniri  =  1  bigoncia,  2£  b.  =  1  migliajo,  2m.= 

1  botta,  for  oil,  ...-==  322.22 

Naples:  GO  carraffa  =  1  barile,  -  -  -  =     11.581 

3£  barile  =  1  salina,  4  s.  =  1  pipa,       -  =  162.137 

12  barile  =  1  botta,  2  botte  =  1  carro.  • 

6  misurella  or  l£  pignata  =  1  quarto,   16  q.  =  1 

stajo,  16  s.  =  lmlma,,  for  oil,  -  =     42.538 

11  salina  =  1  last  for  shipping. 
Sardinia.  —  Genoa :  90  amola  or  5  foglietta  or  pinta  = 

1  barile,  2  b.  =  1  niezzaruola,     -  -  -  =    39.218 

16  quarteroni  =  1  quarto,  4  q.  =  1  barile,/or  oil,  =     17.084 
Turin :  20  quartine  =  1  boccale,  2  b.  =  1  pinta,  6  p. 

=  1  rubbio,  6  r.  =  1  brenta,  10  b.  =  1  carro,      —  148.806 
States  of  the  Church.  —  Ancona: 

.  4  fogliette  =  1  boccale,  24  b.  =  1  barile,         -        =    11.35 

1  soma  of  oil,  -  -  -  -  -=     18.494 

Rome:  Wine  measure:    4  cartocci  =  1  quartuccio,  4 

q.  =  1  foglietta,  4  f .  =  1  boccale,     -  -        =      0.481 

32  boccali  =  1  barile,  16  b.  =  1  botta,       -  -  =  246.544 

Oil    measure:    4  boccali  =  1  cugnatello,    10  c.  =  1 

mastello  or  pello,  2  m.  =  1  soma,  -  -  =»    43.333 

Tuscany.  —  Leghorn,  Florence,  Pisa  : 

4  quartucci  or  2  mezette  =  1  boccale,  40  b.  or  20 
fiasco  =  1  barile  =  133J  libbra  or  99.81  lbs.  Av. 


40  0  FOREIGN   LIQUID   MEASURES  REDUCED  TO  UNITED  STATES. 

Foreign.  U.  States 

"Wine 
gallons 

16  fiasco  =  1  barile,  2  b.  =  1  botta,  for  oil  =  240 
libbra  or  179§  lbs.  U.  S. 
MALTA  I.  —  2  caffisi  =  1  barile  =  50  rotl  or  87|  lbs. 

Av. 
MADEIRA  I.  —  Standard  same  as  Lisbon  (Portugal) . 

M  A I  rEmUS  I.  —  Port  Louis  :  1  velt.        -  -        =       2. 

MEXICO.  — Same  as  Cadiz  (Spain). 

NORWAY.  —  Same  as  Denmark. 

PORTUGAL.  —  Lisbon,  &c.  :  24  quartilhi  or  6  Canada 

=  1  alqueire  or  cantaro,  -  -  -  -  =      2.185 

2  alqueire  =  1  almude,  26  a.  =  1  bota  or  pipa,       =  113.627 
2  bota  =  1  tonelada. 
31  almudes  =  1  pipa,  London  gauge. 
Oporto  :  2  alqueire  =  1  almude,  21  a.  =  1  pipa,  =  139.134 

PRUSSIA  (legal  throughout  the  kingdom  since  1820  :) 

2  ossel  =  1  quart,  30  q.  =  1  anker,  2  a.  =  1  eimer 
=  3840  cubic  Rhein-zolle  or  4192  cubic  inches, 

U.S.,  -  —     18.146 

3  eimer  or  1£  ohm  =  1  oxhoft,  4  o.  =  1  fuder,      -  =  217.758 
3 J  eimer  =  1  fass. 

Dantzie:  3 \  eimer  =  1  fass,  -  -  =     60.487 

6  eimer  =  1  pipe,    -  -  -  -  -  =  108.876 

Konigsberg :  4£  eimer  =  1  pipe,    -  -  =     81.658 

RUSSIA  (legal  for  the  Empire  since  1820)  : 

12£  tscharka  =  1  osmuschka  or  krashka,  2  o.  =  1 

tschet-vverk,  4  t.  =  1  vedro  or  wedro,      -  -  =       3.240 

40  vedro  =  1  botsclika  or  anker,  -  =  129.86 

13J  botschka  =  1  sarokowaja. 
Revel,  Riga:  30  stof  =  5  viertel  =  1  anker,  -  -  =     10.311 

4  ankor  =  1  alim,  6  a.  =  1  fuder,         -  =  247.46 
SARDINIA   I. —  4  quartucci  =  1  quartaro,  8  q.  =  1 

barile,       -  -  -  -  -  -=      8.874 

2  barile  =  1  inezzaruola. 
SICILY    I.  —  Palermo  :    20  quartucci  =  1  quartaro,  8 

<|iiartari  =  1  barile,  -  =       9.430 

12  barile  or  5  salma  =  1  botta  or  pipa,       -  -  =  113.237 

12  salma  =  1  toiinrllata. 

1  nMBOffor  oil**  12|  rotoli  grosso,     -  =»      3.09 

Messina:  12  barile  or  5  salma  =  1  pipa,        -  -  =  108.— 

1  cattBOffor  oil  =  12£  rotoli  groat 
Syracuse:  I  -  salma  =  I  tonnellata,   -  -  -=  247. — 

SPAIN.  —  F"i-  Customs  values,  nee  Cadiz. 

Alicant:    4  copa  =  I    cuartilla,  4   c.  =  1  cuarto,  4 
cuarti  =  1  cantaro  =  1  Castilian  am>l>a  =  25.38 


FOREIGN   LIQUID   MEASURES   REDUCED   TO   UNITED   STATES,     fl  41 


Foreign. 


lbs.  =  3.04  gallons  of  wine  or  3.642  gallons  of  90 
per  cent,  alcohol.  United  State-  measure. 
40  arrobe  =  1  pipa,  2  p.  =  1  toneladft. 
Barcelona:  4  petricon  =  1  mitadclla  or  porrone,  4  m. 
=  1  quartern,  2  q.  »  1  cortan  or  mitjera,  2  e.  = 
1  mallah,  8  m.  =  1  carga  =  12  arrobe  or  205.08 
lbs.  Av. 
4  carga  =  1  pipa. 

4  quarta  =  1  cuarto,  4  c.  =  1  cortan,  30  cortan  = 
1  carga,  for  oil  =11  arrobe  or  243.54  lbs.  Av. 
Cadiz  (Standard  of  Castile)  :  2  copa  =  1  azumbra,  2 
a.  =  1  cuartilla,  4  c.  =  1  cantaro  or  arroba. 
1  arroba  major  =  35  libre  or  35.541  lbs.  Av.,  or 

984|  cubic  inches  of  distilled  water  at  00°, 
1  arroba  menor, /<?;■  oil  =  27.',  libre, 
10  arrobe  =  1  mayo,  27  arrobe  =  1  pipa,  30  arrobe 
=  1  bota,  2  bote  =  1  tonelada. 
Corunna,  Fcrrol:  10  quartilli  =  1  olla,  4  o.  =  1  can- 
ado,  4  c.  =  1  mayo  =  14  arroba  sutil,    - 
Gibraltar :  Same  as  the  United  States. 
Malaga:  8  azumbre  =  1  cantaro  or  arroba  (34J  1.), 
Santander :  8  azumbre  =  1  cantaro  =  20  libre,    - 
Valencia:  1  arroba  =  38  libre  menor, 

1  arroba,  for  oil=  29^  libre  menor, 
SWEDEN.  —  Stockholm,  &c.  :    4  jungfru  or  ort 

quarter,  4  q.  =  1  stop,  2  s.  =  1  kanna,  - 
6  kanne  =  1  atting  or  ottingar,  2  a.  =  1  fjerding, 

1\  f .  =  1  ankare,  2  a.  =  1  embar,  If  e.  =  1 

tunna,  - 

14  tunna  =  1  am,  1£  a.  =  1  oxhufwud, 

2  oxhufwud  =  1  pipa,  2  p.  =  1  fuder, 
SWITZERLAND  (legal  since  1823,  for  the  Cantons  of 

Aarau,  Basle,  Berne,  Freiburg,  Lucerne,  Solothurn, 
Vaud ;  but  not  in  general  use)  : 
10  emine  =  1  mass  or  pot,   10  m.  =  1  gelt  =  lj^y 
decalitres  of  France,  - 

Special  and  local  — 
Basle :  4  schoppen  =  1  mass,  4  m.  =  1  viertel,  2f  v. 

=  1  setier,  1§  s.  =  1  ohm,  3  o.  =  1  saum,  - 
Berne:    8    becher  or  2  viertel  =  1  mass,  25  m.  =  1 
brente  or  eimer,  4  b.  =  1  saum,  - 
'  4  saum  =  1  fass,  1£  f.  =  1  landfass,     - 
Geneva :  2  pot  =  1  quarteron,  24  q.  =  1  setier, 
D* 


U.  States. 

Wine 
gallons. 


4.263 
3.319 


-==    42.533 


=  1 


4.182 
4.739 
3.500 
2.737 


-=      0.089 


33.174 
02.202 
248.81 


=  3.5004 


=  40  337 

44.1G1 

204. 9G6 

11.942 


42  <Z    FOREIGN  LIQUID  MEASURES  REDUCED  TO   UNITED   STATES. 

Foreign.  IT.  States. 

Wine 
gallons. 

12  setier  =  1  char. 
Lausanne :  10  verre  =  1  mass,  10  m.  =  1  broc,  3  b. 

=  1  setier  or  eimer,  -  -  -  -  =     10.G99 

Neufchatel:  8  pote  or  mass  =  1  brochet  or  stutz,  =      4.024 

2£  brochet  =  1  brande,  2§  brande  =  1  gerl,  -  =    26.159 

24  brochet  =  1  muid,  24  m.  =  1  bosse. 
St.  Gall:  lauter-mass  :  8  mass  =  1  viertel,  -        =      2.773 

4  viertel  =  1  eimer,  4e.  =  1  saum,  -  -  =    44.37 

7£  saum  =  1  fuder. 

Schenk-mass :  8  mass  =  1  viertel,         -  =      2.465 

4  viertel  =  1  eimer,  &c. 

8  mass  =  1  viertel,  for  oil,  -  -  -  =      2.867 

Zurich:  lauter-mass:  8  statz  or  2 mass  =  1  kopf,        =     0.9637 
7£  kopf  =  1  viertel,  4  v.  =  1  eimer,  1£  e.  =  1  saum,  =    43.368 
Schenk-mass  :  2  quartli  =  1  mass,  2  m.  =  1  kopf,  =  0.86746 
7£  kopf  =  1  viertel,  &c. 
TRIPOLI  (N.  Africa)  :  14  caraffa  =  1  mataro,  for  oil 
=  42  rotoli  =  46^  lbs.  Av. 
1  barril  =  116|  rotoli. 
TUNIS.  —  2  mettar,/or  wine  =  1  mettar,  for  oil  =  36 
rotoli  =  39T3<j  lbs.  Av. 
1  millerolle  =  120  rotoli. 
TURKEY.  —  1  almud,  =      1.38 

West  Indies. 

In  the  islands  of  Antigua,  the  Bahamas,  Barbadoes,  Bar- 
buda, Dominica,  Grenada,  Les  Saints,  Montserrat, 
Nevis,  St.  Kitts\  St.  Vincent,  Tobago,  Tortola, 
the  pleasures  for  Liquids  are  the  same  as  those  of 
the  United  States,  or  the  same  as  those  of  Great 
Britain,  previous  to  1825. 

In  Jamaica:  85  Imperial  gallons  =  1  puncheon,      -        =  102.03 

In  the  Islands  of  Deseade,  Guadeloupe,  Mariegalante,  Mar- 
tinique, St.  Lucia :  8  muces  =  4  roquilles  =  2 
chopines  =  1  pinte,  8  pintes  =  4  pots  =  2  gallons 

=  1  velt,  -  -  -  -  -  -  to      2. 

35  veltes  =  1  muid. 
This  was  the  system  for  Liquid  Measures  in  France, 
before  1812,  except  that  the  value  of  the  muid 
was  70.855  gallons,  U.  S. 

In  Bonaire,  Curacoa,  Saba,  St.  Eustatius,  St.  Martin  : 
Same  at  Holland  old  measure,  or  measure  before 
1820. 


FOREIGN   LIQUID   MEASURES  SEDUCED  TO   UNITED   STATES,   a  43 
Foreign.  U.  States. 

Wine 
gallons 

In  St.  Bartholomew :  Same  as  Sweden. 

In  Trinidad:  Same  as  Castile  (Spain). 

In  Santa  Cruz,  St.  John,  St.  Thomas :  Same  as  Denmark. 

Cuba  I.  —  Cardenas,   Cienfuegos,    Matanzas,  Nuevitas, 

Porto  Principe,  St.  Jago,  &o. :  Same  as  Castile 

( Spain) . 
Havana:  1  arroba  =  4.1  gallons.     1  bocoy,         -        =»    36.— 
Hayti  I.  — Aux  Cayes,  Cape  Haytien,  Jeremie,  Port  au 

Prince,  Port  Platte,  &c. :  Same  as  France  before 

1812. 
Savanna,  St.  Dmingo,  &c.  :    Same  as  Castile. 
Poeto  Rico  I.  —  Same  as  Castile. 


Ua 


FOREIGN  DRY   MEASURES  REDUCED  TO  UNITED 
STATES. 


Foreign.                                                              U.  States. 

Winchester 
bushels. 

ABYSSINIA.  —  Massuah :  24  madcga  =*  1  ardeb,  -        =*  0.333 

ALGIERS.  —  2  tarrie  =  1  saa  or  aaha,  -            -            -  =  1.125 

8  saa  =  1  caffiso,  -  -   -        -       'em      9. 

100  litres  =  1  hectolitre,     -            -            -            -  =  2.838 

ARABIA.  —  Mocha:  40  kella  =  1  tomaun  (for  rice)  = 
168  lbs.  Av. 

AUSTRIA  (legal  and  general)  : 

2  becher  =  1  fudermassel,  2  f.=  l  muhlmassel,  2 
m.  =  1  achtel,  2  a.  =  1  viertel,  4v.  =  l  metze, 

30  m.  =  1  muth,                                                      =  52.354 
Local  and  special  — 

Trieste:  2  polonic  =  1  metze,  1£  m.  =  1  stajo,          -  =  2.156 
Bohemia.  —  Prague:    12  seidel  =  1  massel,  4  in.  =  1 

viertel,  4  v.  =  1  strich,                                           =  2.656 

3  viertel  =  1  metze,             -            -            -            -=  1.992 
Hungary  :  32  halbe  =  1  viertel,  4  v.  =  1  metze,     -        =  1.745 

Buda  and  Pesth :  1  metze,       -            -            -            -  =  2.-7 

Moravia  :  1  metze,  -            -            -            -                     =  2.— 

AZORE  I.  —  2  meio  =  1  alqueire,  4  a.  =  1  fanga,      -  =  1.359 

BALEARIC   I.  —  Majorca  :  6  barcella  =  1  quartern,     =  2.042 

Minorca:  6  barcella  =  1  quartera,               -                     =  2.156 
BELGIUM.  —  100  uper  =  10  setier  =  3  mudde  =  1 

muid  =  3  hectolitres  of  France,     -            -            -=  8.513 

10  muid  =  1  last,                                                        =  85.134 
Antwerp,  Brussels,  &c.  (old  measures)  — 

Jo  mftlfltar  vat  =  1  halster,  10  h.  =  1  sac,              -  =  6.918 

108  gelte  =  1  muid,                                                     =  8.302 

Ghent :  1  halster,        -             -            -            -            -=  1.4W9 

4  meuke  =  1  raziere,    -                                              =  l>-<» 
Mechlin:    1   Bteake,       -              -              -              -              -=  0.014 

BERM1  DAS  I. — Same  as  United  States. 

BRAZIL.  —  10  quarta  =  1  iun-a,  15  f.  =  1  moio,  -        =  23.02 
I  in  Ian :  1  alqueire,      -            -            -            -            -■■0.863 

Rio  Janeiro :  1  alqueire,                                                =  1.135 


FOREIGN   DRY   MEASURES  REDUCED  TO  UNITED  STATES,      a  45 
Foreign.  U.  States. 

Winchester 
bushels. 

Central  and  South  America. 

Balize,  Campeche,  Nicaragua,  San  Salvador,  Sisal,  &c. 
Buenos  Ay  res,   Callao,  Carthagena,  Laguayra,  Mara- 
caybo,  Montevideo,  Truxillo,  Valparaiso,  &c. :  Samo 
as  Cadiz,  generally. 
Buenos  Ayres :  1  fanega,         -  -  =      3.752 

Montevideo :  1  fanega,       -  -  -        sss      3.868 

Valparaiso :  1  fanega,  -  -  -  -  «=       2.572 

Berbice,  Demerara,  Essequibo,  Surinam:  Same  as  Hol- 
land before  1820. 
Cayenne :  Same  as  France. 
CANARY  ISLANDS.  —  12  celemine  —  1  fanega,  —      1.776 

17  celemine  =  1  fanaga  (heaped) . 
CANADA  EAST.  —  1  minot,  =      1.111 

CANDIA  I.  —  lcarga,  -  -  -  -—      4.323 

CAPE   COLONY.  —  Cape  Town :  4  schepel  =  1  muid, 

10  muid  =  1  load,     ----=:    30.65 
CHINA.  —  By  weight.     See  Weights. 

CORSICA  I.  —  6  bacino  =  1  mezzino,  2  m.  =  1  stajo,  =      4.256 
CYPRESS  I.  — By  weight.     See  Weights.  ' 
DENMARK.  — 4  sextingkar  or  fjerdingkar  =*  1  otting- 
kar  or  skieppe,  2  o.  =  1  fjerding  or  stubchen,  4 
f.  =  1  toende,      -  -  -  -  -=      3.947 

22  toende  =  1  last,  =    86.836 

EGYPT.  —  Alexandria,  Rosetta :  1  kisloz,  -  -  ==      4.85 

24  robi  =  1  rebeb,  =      4.462 

Cairo :  24  robi  =  1  ardeb,      -  -  -  - «       5.165 

FRANCE.  —  100  litres  =  10  decalitres  =  1  hectolitre,   =      2.838 
100  hectolitres  =  10  kilolitres  =  1  myrialitre,        -  =  283.782 
GERMANY  .—  Baden  (legal)  :  1000  becher  or  100  mas- 

sel  or  masslein  or  10  sester  =  1  malter,         -        =       4.256 
10  malter  =  1  zober  =  15  hectolitres  of  France,       -  =    42.567 
Manheim,  Heidelberg :  32  masschel  or  4  immel  or  invel 
or  2  kumpf  or  vierling  =  1  simmer,  2  s.  =  1  vi- 
ernzel,  8  v.  =  1  malter,  for  wheat,     -  =      3.152 

1  malter,  minim,      -  -  -  -  -  =      2.922 

1       "       for  barley  and  oats,     -  -  =       3.546 

Bavaria  (legal)  :  8  masslein  or  4  dreissiger  =  1  achtel 

or  massel,  4  achtel  =  1  viertel,    -  -  -  =      0.526 

12  viertel  =  1  scheffel,  ----=»      6.31 
144  metzen  or  12  mass  =  1  scheffel, /or  oats,  &c,    =       7.363 
4  kubel  =  1  seidel,  6  s.  or  4  scheffel  =  1  muth,/or 

coals  and  lime,      -  -  -  -  -  =    25.24 


46  0     FOREIGN  DRY  MEASURES  REDUCED  TO   UNITED   STATES. 


Foreign. 

Bamberg :  40  gaissil  =  1  simra,  3  s.  =  1  sheffel, 
Bayreuth :  16  mass  =  1  simmer,  - 
Nuremberg :  16  mass  or  2  diethaufe  =  1  metze,  16 
metzen  =  1  malter  or  simmer,     - 
16  hafer-mass  =  1  hafer-metze,  32  hafer-metzen  = 
1  hafer-simmer,  -  - 

Ratisbon:  4  massel  a  1  strich,  - 

1  strich,  for  salt,  &c.,        -  » 

Wurzberg :    144  massel  =  12  mass  =  2  achtel  =  1 
scheffel,    ------ 

1  scheffel, /or  oats,  &c.,  -  -    - 
Hanover  (legal)  :  144  krus  =  24  vierfas  =  18  drittel  or 

metzen  =  6  hhnten  =  1  malter, 
8  malter  =  1  wispel,  2  w.  =  1  last, 
12  malter  =  1  fuder,     - 
Bremen :  16  spirit  =  4  viertel  =  1  scheffel,     - 
40  scheffel  =  1  last,       - 
Hesse  Cassel  :  64  kopfchen  =  32  masschen  =  8  metzen 
=  4  mass  =  2  himten  =  1  scheffel, 

3  scheffel  or  1£  viertel  or  butte  =  1  malter, 
Hesse  Darmstadt  (legal)  :  64  kopfchen  =  32  masschen 

=  8  gescheid  =  2  kumpf  =  1  metze, 

2  metzen  =  1  simmer,  4  s.  =  1  malter, 

4  butte  (coal  measure)  =  1  mass,     -  -  - 
Frankfort:  4  schrott  =  1  mass,  .4  m.  =  1  gescheid,  4 

g.  =  1  sechter,  2  s.  =  1  metze,  2  m.  =  1  simmer, 
4  s.  =  1  achtel  or  malter,       - 
Holstein. —  Hamburg,  Altona:  4  masschen  =  1  spint, 
4  s.  =  1  hinit,  2  h.  =  1  fass,       -  -  - 

20  fass  =  10  scheffel  =  1  wispel, 
4£  wispel  or  1£  last  =  1  stock,         -  -  - 

10  scheffel  (for  barley  and  oats)  =  1  wispel,     - 
45  tonne  (for  coals)  or  30  sacks  =  1  fass,  - 
Lubec:  4  fass  =  1  scheffel,  4  s.  =  1  tonne, 

3  tonne  =  1  dromt,  8  d.  =  1  last,  -  -  - 
96  scheffel  or  24  tonno  =  1  last,  for  oats, 

Kiel:  4  scheffel  =  1  tonne  or  barril,  -  -  - 

Mecklenburg.  —  Rostock,  &c.  :    16  spint  =  4  fass  =  1 
scheffel,  2  s.  =  1  dromt,        - 
2|  dromt  =  1  wispel,  3  w.  =  1  last, 
45  viertel  or  15  stubchen  =  1  drum t,  for  oats,  - 
Saxon  v.  —  Dresden,  Jsipsic:  4  BMWohcn—  1  metze,  4 
m.  =  1  viertel,  4  v.  =  1  sclitflel, 
12  s.  =  1  malter,  2  m.  =  1  wispel, 


U.  States. 

Winchester 
bushels. 

=      6.618 
=    14.044 

-  =      9.028 


16.696 
0.756 
1.51 

5.183 
8.533 

5.296 
84.736 
63.552 

2.021 
80.834 

2.28 
6.482 

0.452 

3.632 

17.736 


=  3.256 

—  1.495 
=  29.892 
=  134.514 
=  44.831 
=  179.6 
=  3.796 
=  91.1 
=  106.918 
=  3.367 

=  13.243 

=  105.944 

=  14.9 

=  2.963 

—  71.11 


FOREIGN   DRY  MEASURES  REDUCED  TO   UNITED  STATUS,      <X  47 

Foreign.  U.  States. 

Winchester 
busheli. 
GREAT  BRITAIN:  Imperial  measure:  See  Dry  Meas- 
ures of  the  United  States  ;  1  bushel,       -  -  =    1.0315 
GREECE.  —  Pair  as :  2  medimni  =  1  staro,            -        =2.23 
lbachel,              -            -            -  =      0.85 
HOLLAND    (legal)  :    10  maatje  =  1  kop,   10  k.  t=  1 
scheppel,  10  s.  =  1  mudde  or  zac,  30  m.  =  1  last 
=  3  kilolitres  of  France,  =    85.134 

India  and  Malaysia  or  East  Indies. 

Birmah.  —  Rangoon :  2  lamyet  =  1  lame,  2  1.  =  1  said, 
4  s.  =  1  pyis,  2   p.  =  1  sarot,  2  s.  =  1  sait,  4 
sait  =  1  ten  or  basket  =  16  vis  =  58.4  lbs.  Av. 
Ceylon  I.  —  Colombo :  24  seers  =  1  parah,        -  -  =      0.721 

Hindostan.  —  Bombay :  Salt  measure  — 

10£  adowlies  =  1  parah,  100  p.  =  1  anna  =  93.033 
cub.  feet,  1G  anna  =  1  rash,  -  -        ==1196. 13 

Grain  measure  —  2  tipprees  =  1  seer,  4  s.  =  1  adoulio, 
16  a.  or  7  pallie  =  1  para  =  186|  lbs.  Av. 
8  para  =  1  candy. 
Calcutta  :  5  chattac  =  1  khoonka,  16  k.  =  1  raik,  4  r. 
=  1  pallie,  20  p.  =  1  soallie  =  154  lbs.  Av. 
8  soallie  =  1  morah  or  maund  bazar. 
16  morah  =  1  kahoon. 
lg  bazar  maunds  =  1  soallie. 
Madras :  8  ollock  =  1  puddy,  8  p.  =  1  marcal,  5  m. 

=  1  para,  -  -  -  =       1.744 

80  para  =  1  garce,  -  -  -  -  -  =  139.535 

Serampore,  Tranquebar  (legal)  :  Same  as  Denmark. 
Tatta :  4  puttoes  =  1  twier,  4  t.  =  1  cossa,  60  c.  = 
1  carvel  =  5£  Tatta  maunds  or  408$  lbs.  Av. 
Java  I.  —  Batavia :  22  mudden  =  1  coyang,  -        =    62.432 

Bantam :  1600  bambou  =  400  gantang  =  52  mudde 

=  1  coyang,  for  rice,        -  -=147.565 

Luzon  I.  —  Manilla:  Same  as  Cadiz  (Spain). 
Malacca.  —  32  mudde  =  1  coyang,  -  -  =    90.81 

Siam.  —  40  sat  =  1  scsti,  40  s.  =  1  cohi,  10  c.  =  1  co- 
yang =  32  hectolitres  of  France,  -  -  -  =    90.81 
Sumatra  I.  — 4  pakha  =  1  culah,  2  c.  =  1  koolah,  15 

koolah  =  1  tub,  =      1.872 

IONIAN  ISLANDS.  —  Corfu,    Paxos:   2  misura  =  1 

bacile,  4  b.  =  1  moggio,  -  -  -  -  =      4.777 

Cephalonia :  8  misure  =  4  bacile  =  1  moggio,     -        =       5.6 
Zante:  8  misure  or  4  bacile  =  1  moggio,       -  -  =      5.116 


48  a    FOREIGN   DRY   MEASURES   SEDUCED   70   UNITED   STATES. 

Foreign.  U.  States. 

Winchester 
bushels. 

Ithaca:  5  bacile  =*  1  moggio,  -  -  -  -=      5. 

ITALY. — Lombardy   and    Venice. —  Government   and 
Customs  Measure:  10  coppi  =  1  pinta,  10  p.  = 

1  mina,  10  m.  =  1  soma  =  1  hectolitre  of  France,  =      2.838 
Special  and  local  — 

Venice :  4  quartaroli  =  1  quarto,  4  q.  =  1  stajo  or 

staro,  4  8.  =  1  moggio,  -  -  =       9.08 

Naples.  —  3  misura  =  1  stopello,  4  s.  =  1  mezetta,  2  m. 

=  1  tomolo,  36  t,  =  1  carro,       -  -  -  =    54.81 

Sardinia. —  Genoa:  12  gombetti  —  1  ottaro  or  quarto, 

8  quarto  =  1  mina,  -  =      3.426 

8  mine  =  1  niondino, /or  salt. 
Nice:  4  motureau  =  1  quartier,  4  q.  =  1  stajo,  4  staji 

=  1  sacco,  -  -  -  -  -  =      3.405 

Turin  :  20  cucchiari  =  1  copello,  4  c.  =  1  quartiere, 

2  q.  =  1  mina,  2  m.  =  1  stajo,  3  s.  =  1  6acco,     =      3.263 
States  of  the  Church.  —  Ancona : 

4  provenda  =  1  coppa  or  lappa,  2  c.  =  1  corba,        =       2.03 
Rome:  5£  quartucci  or  lg  scorzi  =  1  starello,  2  starelli 

or  1£  staji  =  1  quartarello,  -  -  -  =       1.044 

4  quartarelli  or  2  quarte  =  1  rubbiatilla,  2  rubbia- 

tille  =  1  rubbio,        -  -  =      8.356 

4£  rubbi  =  1  tonnellata  {shipping) . 
Tuscany.  —  Florence,  Leghorn,  Pisa : 

8  bussole  =  4  quartucci  =  2  mezette  =  1  metadella, 
4  m.  =  1  quarto,  4  q.  or  2  mina  =  1  stajo,  3  s. 
=  1  sacco,  8  sacci  =  1  maggio,   -  -  -  =    16.592 

JAPAN.  —  10  gantang  =  1  ickoga,  100  ickoga  =  1 

icmagoga,  100  icmagoga  =  1  managoga. 
MALTA.  — 1  salma  (rasa),  =      8.22 

1  sidma  (colina),     -  -  -  -  =      9.56 

MADEIRA   I.  — Standard  same  as  Lisbon  (Portugal). 
MEXICO.  — Same  as  Cadiz  (Spain). 

MOROCCO.  —  Mogadore:  1  mud,  -  -  -        =*       5.184 

NORWAY.  —  Same  as  Denmark. 
PORTUGAL.  —  Lisbon,  St.  Ubes,  &c.  : 

16  quarto  =  8  meio  =  4  alqueire  =  1  fanga,  -  =       1.534 

15  boga  =  I  majo,  4  m.  =  1  lust,       -  -       =*    92.087 

27  fanga  =  1  toneladft,  for  shipping. 

1  l>:ilde,  for  coals,     -  -  -  -  -=     12.69 

1  fanga,  "      "  ...  =     21.167 

Oporto  :  1  fanga  =  1.937  bus.    1  raze,  for  salt,  =      1.25 

PRUSSIA  (legal  since  1820)  :  4  masschen  ==  1  metzo,  4 


FORJBHJN   DRY    MEASURES   REDUCED  TO   UNITED  STATES,     a  49 

Foreign.  U.  States. 

Winchester 
bushels. 

metze  =  1  viertel,  4  v.  =  1  scheffel  =  3072  cubic 
Rhein  zolle  or  3353f  cubic  inches,  U.  S.,  -  -  =      1.559 

12  scheffel  =  1  malter  or  dromt,  2  in.  =  1  wispel,    =    37.431 

3  wispel  =  1  last 

2  wispel  =  1  last,  for  barley  and  oats. 
RUSSIA  (legal for  the  Empire)  : 

8  garnetz  =  1  tschetwerik,  2  t.  =  1  payak,       -        =       1.438 
2  p.  =  1  osmin,  2  o.  =  1  tschetwerk,  -  -  =      5.952 

14  tschetwerk  =  1  kuhl,  16  tschetwerk  =  1  last. 
Libau,  Revel,  Riga :  12  stof  =  1  kulmet,  3  k.  s=  1  lof, 
24  1.  =  1  tonne,  2  t.  =  1  last. 
SARDINIA   I.  —  Cagliari,  &c.  :  4  imbuto  =  1  carbula, 

4  c.  =  1  starello,  3  s.  =  1  restiere  or  rasiera,        =      4.166 
SICILY  I.  —  Palermo,  Messina :  2  stari  =  1  modello,  4 

m.  =  1  tomolo,  4  t.  =  1  bisaccia,  4  b.  =  1  salma,  =      7.81 
16  tomoli  grosso  =  1  sahna  grosso,        -  -        =9.72 

SPAIN.  —  Alicant :  2  medio  =  1  celemin,  4  c.  =  1  bar- 

cella,  12  b.  =  1  cahiz,    -  -  -  -  =      6.992 

Barcelona :  4  picolin  =  1  cortain,  12  c.  =  1  quartera, 

2£  q.  =  1  carga,  if  c.  =  1  salma,    -  =      8.191 

Cadiz  (Standard  of  Castile)  :  2  medio  =  1  celemin,  12 

c.  =  1  fanega,  12  f.  =  1  cahiz,    -  -  -  <-»    19.189 

4  cahiz  =  1  last,  =*    76.759 
Corunna,  Ferrol:  4  celemine  =  1  ferrado,  3  f .  =  1 

fanega,  12  fanega  =  1  cahiz,       -  -  -  =*    19.189 

Gibraltar :  Same  as  Cadiz. 
Malaga  :  Same  as  Cadiz. 

Santander :  144  celemin  =  12  fanega  =  1  cahiz    -        =    24.984 
Valencia :  8  medio  =  4  celemin  =  1  barchilla,  12  bar- 

chille  =  1  cahiz,  -  -  -  -  -  =      5.758 

SWEDEN.  —  Stockholm,  &c.  :  2  stop  =  1  kanna,  11  k. 
=  1  kappe,  4  kappe  =  1  fjerding,  4  f .  =  1  spann, 
2  spann  =  1  tunna,  =      4.158 

24  tunne  =  1  last. 
SWITZERLAND  (legal,  since.  1823,  for  the  Cantons  of 
Aarau,  Basle,  Berne,  Freiburg,  Lucerne,  Solothurnt 
Vaud ;  but  not  in  general  use)  : 
10  emine  =  1  gelt  or  quarteron,  10  g.  =  1  sac  = 

L/tj-  hectolitres  of  France,  -  -  -  -=      3.831 

Special  and  local  — 
Basle :  2  bacher  =  1  kopflein,  8  k.  =  1  sester,     -        =       0.97 

4  sester  =  1  sack,  2  s.  =  1  vierzel,  -  -  -  =       7.756 

Berne :  4  achterli  or  2  immi  =  1  massli,  2  m.  =  1 

mass,  12  mass  =  1  mut,         -  -  =      4.771 

E 


50  a    J OREIGN   DRY  MEASURES  REDUCED  TO  UNITED  STATES. 

Foreign.  U.  States. 

Winchester 
bushels. 

Geneva :  2  bichet  =  1  coupe  or  sac,    -            -  -  —  2.203 

Lausanne :  10  copet  =  1  emine,  10  e.  =  1  sac,  -        =  3.831 

Neufchatel :  3  copet  =  1  emine,  8  e.  =  1  sac,  -  -  =  3.459 

3  sacks  =  1  muid. 

St.  Gall :  4  massli  =  1  vierling,  4  v.  =  1  viertel,  4 

viertel  =  1  mutt,  2  m.  =  1  malter,     -  =    4.688 

Zurich  :  16  massli  or  4  vierlmg  =  1  viertel,  4  v.  =  1 

mutt,  4  m.  =  1  malter,    -  -  -  -  =      9.333 

1  mass,  for  salt,  =      2.622 

4  mass  =  1  korb. 

TRIPOLI  (N.  Africa)  :  20  tiberi  =  1  caffiso,    -  -  =  1.15*4 

4  orbah  =  1  tomen,  3 J  t.  =  1  nusfiah,  -        =  1.0157 
3  n.  =  1  ueba. 

TUNIS.  —  12  zah  or  saha  =-  1  quiba,  16  q.  =  1  caffiso 

=  18  wage,          -            -            -            -  -  =  14.954 

TURKEY.  —  Constantinople :  4  kibz  =  1  fortin,  -        =  3.764 

Latakia,  Aleppo :  1  garave,      -            -            -  -=  41.15 

Smyrna:  4  kilo  =  1  fortin,  =  5.824 

West  Indies. 

In  the  Islands  of  Antigua,  the  Bahamas,  Barbadoes,  Bar- 
buda, Dominica,  Grenada,  Jamaica,  Les  Saints, 
Montserrat,  Nevis,  St.  Kitts\  St.  Vincent,  Tobago, 
Tor  tola,  the  Dry  Measures  are  the  same  as  those 
of  the  United  States. 

In  Deseade,  Guadeloupe,  Mariegalante,  Martinique,  St. 
Lucia :  3  boisseau  =  1  minot,  2  m.  =  1  mine,  12 
mines  =  1  setier,  2  s.  =  1  muid,  -  -  =    53.153 

This  being  the  system  of  France  before  1812. 

In  Bonaire,  Curacoa,  Saba,  St.  Eustatius,  St.  Martin: 
Same  as  in  Holland  before  1820  :  See  Holland. 

In  Santa  Cruz,  St.  John,  St.  Thomas :  Same  as  Denmark. 

In  St.  Bartholomew :  Same  as  Swkdkx. 

In  Trinidad:  Same  as  Castile  (Spain). 

Cuba  I.  —  Same  as  Castile,  generally. 

Havana:  4  arrobas  =  1  fanega,  ...  .  =      3.114 

Hayti  I.  —  Aux  Cayes,  Cape  Haytien,  Port  au  Prince, 
Port  Platte,  &c. :  Same  as  France  before  1812. 
Savanna,  St.  Domingo,  &c.  :  Same  as  Castile. 

Porto  Rico  I.  —  Same  as  Castile  (Spain). 


CUSTOM  HOUSE  ALLOWANCES  ON  DUTIABLE  GOODS.  a  51 


CUSTOM  HOUSE  ALLOWANCES  ON  DUTIABLE  GOODS. 

Draft,  or  Tret,  is  an  allowance  of  weight  for  supposed  waste  on 
articles  paying  duty  by  the  pound.  It  is  deducted  from  the  actual 
gross  weight  of  the  article,  and  is  established  as  follows :  — 

Cv/t.  Cwt.  Draft. 

.   On         1  (112  lbs.)  lib. 

Above  1  and  under  .         .         2,    .   2  " 

On        2  "       "        .         .         .         3,    .   3  " 
"  3  "       "        .         .         .       10,    .   4  " 

10  "       "        .         .         .       18,    .    7  " 
"         18  "    upwards,  9  " 

Tare  is  the  weight — actual  or  assumed  by  law  —  of  the  cask, 
sack,  &c,  in  which  the  article  paying  duty  is  contained.  It  is  de- 
ducted from  the  actual  gross  weight  less  the  draft.  The  remainder 
is  the  net  weight  on  which  the  duty  is  assessed,  and  the  weight  at 
which  the  heavy  purchasers  receive  the  goods. 

Leakage  is  an  allowance  on  the  gauge  of  molasses,  oils,  wines,  and 
ail  liquids  in  casks.  It  is  established  at  2  per  cent.,  and  is  deducted 
from  the  actual  gross  gauge,  less  the  real  wants  of  the  cask. 

Breakage  is  an  allowance  of  10  per  cent,  on  ale,  beer  and  porter,  in 
bottles,  and  5  per  cent,  on  all  other  liquors  in  bottles  ;  or,  if  the 
importer  prefer,  the  duties  are  assessed  by  actual  count,  he  so  electing 
at  the  time  of  making  the  entry.  Common  sized  bottles  are  computed 
to  contain  1\  gallons  per  dozen. 

On  bottles  in  which  wine  is  imported  there  is  assessed  a  duty  of 
two  dollars  per  gross,  in  addition  to  the  duty  on  the  wine. 

The  following  articles,  whether  intended  for  sale  or  otherwise,  are 
admitted  into  the  United  States,  from  foreign  ports,  free  of  duty  ;  but 
nevertheless  must  pass  through  the  Custom  House  in  manner  the 
same  as  goods  on  which  a  duty  is  assessed.  • 


Animals  imported  for  breed. 

Antiquities. 

Bulbs  or  bulbous  roots. 

Bullion,  silver  or  gold. 

Canary  seed. 

Cardamon  seed. 

Coins,  gold,  silver,  or  copper. 

Copper  sheathing,  14  by  48  inch, 

and  from  14  oz.  to  34  oz.  per 

square  foot. 
Copper  ore. 
Cotton. 


Cummin  seed. 
Fossils. 
Gold  dust. 
Guano. 

Gypsum,  unground. 
Oakum  and  old  junk. 
Oysters. 

Platina,  unmanufactured. 
Silver,  old,  fit  only  for  re-manu- 
facturing. 
Vanilla,  plant  of. 


52  a  CT7ST0M  HOUSE  ALLOWANCES  ON  DUTIABLE  GOODS, 


TABLE    OF    ESTABLISHED    TARES. 


(a=by custom;  c=Iegal. 
Almonds,  in  bags, 
Alum,  casks, 
Beef,  jerked,  drums, 
"       hhds., 
Bristles,  Archangel, 

"  Cronstadt, 

Camphor,  crude,  tubs, 
Candles,  boxes, 

"  chests,  160  lbs. 

Candy,  sugar,  baskets, 

"  "       boxes, 

Cheese,  hps.  or  baskets, 

"        boxes, 
Chocolate,  boxes, 
Cinnamon,  mats, 

"  chests, 

Cloves,  casks, 
Cocoa,  bags,  {actual  2) 

"       casks, 

"       zeioons, 
Coffee,  E.I. ,  grass  bags, 

"         "      bales, 

"         "      casks, 

"      W.I.,bags, 
Copperas,  casks, 
Cordage,  lines,  bales, 
Cordage,  mats, 
Corks,  bales,  light, 

11       heavy, 
Cotton,  bales, 

"  zeroons, 
Currants,  casks, 
Figs,  boxes,  60  lbs., 

"     £  "       30    " 

t<  li         15     m 

"     drums, 

"  frails,  75  lbs., 
Glue,  Russia,  boxes, 
Indigo,  bags  or  mats, 

"       barrels, 


Per 

lb*. 

cent. 

per 

a  4 

Pkg- 

Indigo,  casks, 

alO 

"        zeroons. 

70 

Looking-glasses,  Fr., 

112 

Mace,  kegs, 

all 

Nails,  casks, 

al2 

Nutmegs,  " 

Ochre,  French,  casks, 

a35 

c  8 

Pepper,  bales, 

«20 

"         bags, 

a  5 

M        casks, 

clO 

Pimento,  bales, 

«10 

"         bags, 

«20 

"         casks, 

clO 

Prunes,  boxes, 

alO 

Raisins,  Malaga,  boxes, 

16 

"           "          casks, 

al2 

jars, 

c  I 

M       Smyrna,  casks, 

clO 

Salts,  glauber,  casks, 

a  8 

Shot,  casks, 

2 

Soap,  French,  boxes. 

c  3 

"      boxes,       (a  more) 

cl2 

Steel,  bundles, 

c  2 

"     cases, 

alO 

Sugar,  bags  or  mats, 

c  3 

11       boxes, 

a\\ 

"       casks, 

5 

"       canisters, 

«20 

"       Java,  willow    > 
baskets,        J 

c  2 

c  6 

Tallow,  casks, 

«12 

"         zeroons, 

alO 
a  6 

Tea,  caddies,  Kctaa,< 
li    «"»*».    ) weight.} 

alO 

Twine,  bales, 

a  4 

11         casks, 

al5 

Wool,  Germany,  bale, 

c  3 

"       S.  Amer.,  bale, 

cl2 

"      Smyrna,       " 

«15 

YB  18482 


284302 


UNIVERSITY  OF  CALIFORNIA  LIBRARY 


